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TRIGONOMETRY UNIT NOTES PART 1 1 Precalculus Pretest- Trig Name:____________________________________ Date:_____________________________________ 1. Given triangle ABC with hypotenuse AB. If AC = 5 and BC = 9, find: a. AB b. Angle A 2. A man stands on the top of a 1000 foot building. He notices a car on the ground and an angle of depression of 32 degrees. How far away from the base of the building is the car? 3. A submarine uses SONAR to detect the passage of other vessels. A submarine detects an enemy destroyer on the surface of the ocean at an angle of elevation of 61 degrees. In order to target the destroyer, the sub needs to determine the range to the target (the straight line distance between the two vessels). If the sub is traveling at a depth of 650 feet, what is the range of the target? 4. Find cos 210 without using a calculator. 2 Precalculus Lesson- Intro to Trigonometry Name:____________________________________ Date:_____________________________________ Objective: To review special right triangles and learn the basics about trig ratios. DO NOW: 1) Find the measure of the hypotenuse of a right triangle if the legs are 1 and 2. 2) Given equilateral triangle ABC, find the length of the altitude to AB if BC=4. 3) What is the length of the diagonal of square ABCD if AB=5? __________________________________________________________________________________________ The Three Basic Trig Functions ex: Find the length of the missing side of the triangle and the exact value of the three trigonometric functions of the angle theta ( ) in the figures below: 1) 5 12 2) 7 11 3 3) 3 4) 2 5) Find the values of the three trigonometric functions for angle coordinates (-3, -5) lies on its terminal side. 7 1 in standard position if a point with the 4 Trigonometric Function Origin Timeline Hindu text Aryabhatiya of Aryabhata is written and refers to a half-chord known as ardhajya Ca. 510 a.d. ardha-jya or half chord is turned around to jya-ardha (“chordhalf”) which in due time is shortened to jya or jiva (or bowstring). Edmund Gunter coins the term: cotangens, which later becomes cotangent or “complement of tangent” The first printed table of secants appeared in the work Canon doctrinae triangulorum By Leipzig Ca. 1114-1187 a.d. Aryabhatiya is translated from Hindu to Arabic. Jiva becomes Jiba which becomes Jayb. When translated into Latin from Arabic, Jayb (bosom of a garment) becomes Sinus (meaning a bend or gulf, or the bosom of a garment). Anatomical meaning: cavities or bays in the facial bones 1551 1583 Tangent is translated from Latin Tangere, to touch. 1585-95 1620 Sinus is translated to Sine in English which means curve, fold, pocket. Geometric definition established: a line drawn from one extreme of an arc of a circle to the diameter that passes through its other extremity. Sir Jonas Moore introduces “Cos” as the abbreviation for “cosinus” 1658 Edmund Gunter coins the term cosinus which means “complement of sinus” 1674 1700 The term “Cosecant:” complement of secant is coined 5 Precalculus Activity- Discovering trig functions Objective: To learn the origin of trig functions. DO NOW: What are the 3 trig ratios? Name:__________________________________________ Date:___________________________________________ Follow the directions below and be sure to round each answer to the nearest ten-thousandth. 1. Each person in your group should have a set of 3 triangles: a. 45-45-90 b. 30-60-90 c. 53-37-90 2. Measure the dimensions of each 45-45-90 right triangle (triangle 1) and write the results in the table provided. 3. Create sine, cosine, and tangent ratios for your 45-45-90 right triangle (results should be rounded to the nearest ten-thousandth) 4. Average the corresponding ratios with the other members in your group. 5. Repeat the process using the 30-60-90 triangle using the 30 degree angle as reference (triangle 2) 6. Then repeat the process using the 60 degree angle as a reference (also triangle 2 but from a different perspective) 7. Finally, repeat the process using the 53-37-90 triangle (triangle 3) 6 Angle 45 1 30 2 60 Also 2 53 3 Length of Horizontal Leg Length of Vertical Leg Length of Hypotenuse Your Ratios (4 decimals) Group Average Ratios Class Average Ratios Sin 45= Sin 45= Sin 45= Cos 45= Cos 45= Cos 45= Tan 45= Tan 45= Tan 45= Sin 30= Sin 30= Sin 30= Cos 30= Cos 30= Cos 30= Tan 30= Tan 30= Tan 30= Sin 60= Sin 60= Sin 60= Cos 60= Cos 60= Cos 60= Tan 60= Tan 60= Tan 60= Sin 53= Sin 53= Sin 53= Cos 53= Cos 53= Cos 53= Tan 53= Tan 53= Tan 53= 7 Follow Up Questions 1. What did we accomplish by averaging the corresponding ratios? 2. Will this process work for the sine, cosine, tangent for an angle of any measure? Why/why not? 3. Find the percent difference from the observed class averages and actual values of sine, cosine, and tangent for each of the angles? Observed Actual 100 Formula: Actual Example: Let’s say my class determined the sine of 45 degrees to be 0.7273. But, my calculator says it should be 0.7071. I would take my answer and subtract from it the actual answer and then divide by the actual answer. .7273 .7071 100 2.8567% .7071 That tells me that my answer was off by 2.8567% for sine. So I would enter that in the appropriate box. So… Angle 45 % Difference for Sine % Difference for Cosine % Difference for Tangent 30 60 53 4. Are there significant percent differences between the Observed and Actual trig ratios? What factors might cause such percent variations? 8 Precalculus Homework- Discovering trig functions Name:__________________________________________ Date:___________________________________________ Create a set of 4 similar triangles (triangles that have the same angles but are different sizes), on separate paper, with the same reference angle (any angle except the ones used in this activity) and complete the table. You will need a protractor. Triangle Measure of Horizontal Leg Measure of Vertical Leg Measure of Hypotenuse Your Ratios (4 decimals) Sin___= Cos___= Tan___= Sin___= Cos___= Tan___= Sin___= Cos___= Tan___= Sin___= Cos___= Tan___= Average Ratios: Sin___= Cos___= Tan___= 9 Precalculus Activity- What is a radian? Name:____________________________________ Date:_____________________________________ Objective: To discover what a radian is. Follow the directions below and be sure to round each answer to the nearest ten-thousandth. Do all work on separate paper. 1. Use the paper provided to cut as many horizontal strips as you can. For convenience purposes, make sure each is about ¾ inch wide. 2. Use a piece of tape to tape the edges together to form a quasi-cylinder. 3. Trace the circular edge on another sheet of paper and estimate the center. 4. Measure the distance from the center to the circumference of the circle. 5. Unfurl the paper and measure its length. 6. Determine how many times the length of the radius goes into the distance found in step 5. 7. Make a table comparing your width of paper, distance from center to edge, and quotient. 8. Find the mean of all of your quotients. What does the mean represent? 9. How many radians are in the circumference of a circle? 10. 1 radian is approximately equal to ______________ degrees. 11. There are ________ radians in 180 degrees and there are ________ radians in 360 degrees. 12. In your own words, a radian is…? PRACTICE: As you do the practice problems, see if an equation for the conversion from radians to degrees and degrees to radians becomes apparent. Find the radian measure for each degree measure: Find the degree measure for each radian measure: 1. 720 1. 2. 90 2. 3. 45 3. 4. 60 4. 2 2 6 2 3 10 Precalculus Lesson: The building blocks of trig functions Name:____________________________________ Date:_____________________________________ Building Blocks of the Unit Circle Graph set up: Quadrantal Angles: Coterminal Angles: Examples: Find an angle that is COTERMINAL with each. *Note: it is often helpful to draw a diagram when solving these types of problems. 1. 100º 2. 650º 3. 405º 4. 400º 11 Reference Angles: Examples: Find an angle that is the REFERENCE ANGLE of each. *Note: it is often helpful to draw a diagram when solving these types of problems. 5. 100º 6. 650º 7. 405º 8. 400º Function of a positive acute angle: Examples: Express the given function as a function of a positive acute angle. 1. tan 225º 2. cos 100º 3. cos 405º 4. 6. cos 400º sin 650 º 5. tan (-120º) 12 Precalculus 13 Precalculus 14 Find the exact value of each expression without using a calculator: 1. 2. tan 135 sin 150 3. 4. cos 210 cos 315 Practice: Express the given function as a function of a positive acute angle and, if possible, find the exact function value. 5. tan 225º 6. cos 100º 7. cos 405º 8. sin 650º 9. tan (-120º) Find the exact value of the given expression. 11. tan 135º + sin 330º 12. sin 300º + sin (-240º) 10. 13. cos 400º (sin 60º)(cos 150º) – tan (-45º) 15 Precalculus Review- Basic Trig Quiz Name:____________________________________ Date:_____________________________________ Objective: To review the following concepts in preparation for a test I. SOHCAHTOA II. Coterminal, reference, and quadrantals III. Finding exact trig values IV. Unit Circle MIXED PBLM SET Answer each of the following neatly and completely and show all work. NO CALCULATORS. 1. Find the exact value of each expression: a. cos 270º b. if cos 7 and sin 8 3. Without finding , find the exact value of tan 4. Find the values of the three trigonometric functions for angle coordinates (-3, -5) lies on its terminal side. 5. Given the following triangle find the measure of angle 3 sin 90º 0. in standard position if a point with the exactly 6 16 8 and tan 9 if sin 6. Without finding , find the exact value of cos 7. Find the values of the three trigonometric functions for angle the coordinates (5, -4) lies on its terminal side. 8. in standard position if a point with Find the length of the missing side and the exact value of the three trigonometric functions of the angle in each figure: a) b) 8 2 12 d) 9 7 7 e) 13 c) 3 5 9. 0. 11 Find the exact value of each expression without using a calculator: a. b. sin 150 cos 210 c. cos 315 17 Scrap Page 18 Deg 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 Rad 1 6 1 4 2 6 4 6 3 4 5 6 6 , 6 4 4 7 6 5 4 8 6 9 , 6 6 4 10 6 7 4 11 6 1 2 0 2 12 6 , 8 4 0 2 Sin 0 2 1 2 2 2 3 2 3 6 , 2 4 4 2 Cos 4 2 3 2 2 2 1 2 0 2 1 2 2 2 3 2 4 2 Tan 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 2 2 Csc Sec Cot 0 6 3 2 2 2 2 2 2 0 1 2 2 2 2 2 2 2 2 2 3 4 2 3 1 2 2 2 3 2 3 2 2 2 1 2 4 2 0 2 3 2 1 2 2 2 1 2 3 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 1 2 3 4 3 2 1 0 2 2 2 2 2 2 2 3 2 1 0 1 2 2 2 4 3 2 1 0 1 2 3 4 3 4 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 19 Precalculus Lesson- Reciprocal Trig Functions Name:____________________________________ Date:_____________________________________ Objective: To learn about the reciprocal trig functions csc, sec, cot. DO NOW: Construct the unit circle in the space provided. __________________________________________________________________________________________ Reciprocal Trig Functions Examples: Find the exact values of the following trig functions. 1. sec 300 2. cot 270 3. 4. 6. 5. (se csc (-210) 20 Precalculus Lesson: The Wrapping Function Name:____________________________________ Date:_____________________________________ Objective: Use radians to solve trig functions DO NOW: Determine the exact value of sin (-45) without using the calculator. Sketch a figure and find the coordinates for each circular point on the unit circle: 5 8 1. 2. 3 6 3. 7 6 Find the sine, cosine, and tangent of each radian measure: 3 5. 6. 2 4 4. 11 3 7. 2 21 Precalculus Wkst: The Wrapping Function Name:____________________________________ Date:_____________________________________ 22 Precalculus Lesson: Radians and Arc Length Name:____________________________________ Date:_____________________________________ Objective: Discuss the relationship among central angles, radii and arc lengths. DO NOW: Determine the exact value of tan (-45) without using the calculator. __________________________________________________________________________________________ What is a central angle? What is an arc length? Relationship among central angle, radius and arc length: Examples: 1. Find the measure of a positive central angle that intercepts an arc of 14 cm on a circle of radius 5 cm. 2. Find the length of the arc intercepted by a central angle of 3.5 radians on a circle of radius 6 m. 3. A wheel of radius 18 cm is rotating at a rate of 90 revolutions per minute. a. How many radians per minute is this? b. How many radians per second is this? c. How far does a point on the rim of the wheel travel in one second? d. Find the speed of a point on the rim of the wheel in centimeters per second. 23 Precalculus Lesson: Working with central angles, radii and Arc lengths Objective: Name:_______________________________ Date:________________________________ find the length of an arc given the measure of the central angle find the radian measure of a central angle given an arc and the radius (1) Find the measure of a central angle (2) Given a central angle of opposite an arc of 24 cm in a circle with a radius of 4 cm. 2 , find the length of its intercepted arc in a circle of radius 14 cm, rounding to 3 the nearest tenth. 24 (3) Given a central angle of 125 , find the length of its intercepted arc in a circle of radius 7 feet, rounding to the nearest tenth. (4) An arc is 14.2 cm long and is intercepted by a central angle of 60 . Rounding to the nearest tenth, what is the radius of the circle? (5) The diameter of a circle is 22 inches. If a central angle measures 78 , find the length of the intercepted arc to the nearest tenth. (6) If the pendulum of a grandfather clock is 44 in long and swings through an arc of 6 , find the length, to the nearest tenth of an inch, of the arc that the pendulum traces. 25 Precalculus HW- Working with radian measure Name:_____________________________ Date:______________________________ SHOW ALL WORK: (1) Find the exact radian measure of a central angle (2) Given a central angle of 128 , find the length of its intercepted arc in a circle of radius 5 centimeters. (Round to the nearest tenth.) (3) An arc is 12 yards long and is intercepted by a central angle of opposite an arc of 30 feet in a circle of radius 12 feet. 3 radians. Find the radius of the circle. 11 (Round to the nearest tenth.) (4) Find the exact radian measure of a central angle meters. opposite an arc of 27 meters in a circle of radius 18 26 SHOW ALL WORK: (5) Given a central angle of 147 , find the length of its intercepted arc in a circle of radius 10 meters. (Round to the nearest tenth.) (6) An arc is 70.7 meters long and is intercepted by a central angle of 5 radians. Find the diameter of the 4 circle. (Round to the nearest tenth.) (7) The figure below shows a stretch of roadway where the curves are arcs of circles. Find the length of the road from point A to point E. (Round to the nearest hundredth.) 0.70 mi D C A 1.8 mi 80 84.5 1.46 mi B 0.67 mi E 27 Precalculus Lesson/HW: Real world trig problems Name:_______________________________ Date:________________________________ Objective: solve real-world applications using trigonometric functions (1) Pizza is typically measured by its diameter. Steve orders a 14 inch pie and cuts it into six equal slices. Find the length of the crust of each slice to the nearest tenth of an inch. (2) Use the diagram to the right to answer the following two questions: (a) Find the angle measure in radians created by the two hands of this clock at 5:00. (b) The minute hand of this clock is 12 centimeters long. Find how far the tip of this hand moves in 10 minutes. (Round to the nearest tenth.) (3) A belt connects a pulley of 2-inch radius with a pulley of 5-inch radius. If the larger pulley turns through 10 radians, through how many radians will the smaller pulley turn? (4) Through how many radians does a pulley of 10-centimeter diameter turn when 10 meters of rope is pulled through it without slippage? 28 (5) A wheel has a radius of 2 feet. As it turns, a cable connected to a box winds onto the wheel. (a) (b) 225 How far (to the nearest tenth) does the box move if the wheel turns 225 in a counterclockwise direction? Find the number of degrees (to the nearest tenth) the wheel must be rotated to move the box 5 feet. BOX 2 ft (6) Through how many radians does a pulley of 6-inch diameter turn when 4 feet of rope is pulled through it without slippage? (7) Two gears are interconnected. The smaller gear has a radius of 2 inches, and the larger gear has a radius of 8 inches. The smaller gear rotates 330 . Through how many radians (to the nearest tenth) does the larger gear rotate? 29 30 31 32 ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH: (4) The adjacent sides of a parallelogram measure 14 centimeters and 20 centimeters, and one angle measures 57 . Find the area of the parallelogram. (5) The base of an isosceles triangle is 48.8 ft long and its vertex angle measures 38.6 . Find the length of each leg. (6) A small rectangular park is crossed by two diagonal paths, each 280 m long, that intersect at a 34 angle. Find the dimensions of the park. 33 Precalculus Lesson- More geometric applications of trig Name:_______________________________ Date:________________________________ Objective: solve geometric applications using trigonometric functions ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH: (1) If a regular pentagon is inscribed in a circle of radius 5.35 centimeters, find the length of one side of the pentagon. (2) If a circle of radius 4 feet has a chord of length 3 feet, find the central angle that is opposite this chord to the nearest degree. 34 ANSWER THE FOLLOWING QUESTIONS SHOWING ALL WORK AND ROUND TO THE NEAREST TENTH: (3) Find the perimeter of a square inscribed in a circle of radius 5 centimeters. (4) The sides of a parallelogram are 20 centimeters and 32 centimeters long. If the longer diagonal measures 40 centimeters, find the measures of the angles of the parallelogram. (5) Each base angle of an isosceles triangle measures 42 30 . The base is 14.6 meters long. (a) Find the length of a leg of the triangle. (b) Find the altitude of the triangle. (c) Find the area of the triangle. 35 Trig Unit Part 1 Definitions, Laws & Formulas The following trigonometric identities hold for all values of expression is defined: Reciprocal Identities where each sin 1 csc cos 1 sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan The following trigonometric identities hold for all values of Quotient expression is defined: Identities sin cos tan sin cos where each cot Two angles in standard position that have the same terminal side. If is the degree Coterminal measure of an angle, then all angles measuring + 360k , where k is an integer, Angles are coterminal with . Since angles differing in degree measure by multiples of 360 are equivalent, every angle has infinitely many coterminal angles. A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. So, for any angle , 0 < < 360 , its reference angle is defined by: (a) , when the terminal side is in Quadrant I (b) 180 – , when the terminal side is in Quadrant II Reference Angles (c) – 180 , when the terminal side is in Quadrant III (d) 360 – , when the terminal side is in Quadrant IV If the measure of is greater than 360 or less than 0 , it can be associated with a coterminal angle of positive measure between 0 and 360 . This special relationship shows that cofunctions have equivalent values for complementary angles: sin = cos (90 – ) cos = sin (90 – ) Cofuntions tan = cot (90 – ) cot = tan (90 – ) sec = csc (90 – ) csc = sec (90 – ) Degree & Radian 1 radian Conversion Formulas 1 deg ree 180 deg rees (or about 57.3 ) 180 radians (or about 0.017 radian) Coordinates of for any points (x, y) on the unit circle, given angle , W( ) = (cos , sin ) Circular Points 36 For any angle in standard position with measure , a point P(x, y) on its terminal Trigonometric Functions of an Angle in Standard Position side, and r sin csc x2 y r r y y 2 , the trigonometric functions of are as follows: cos x r tan sec r x cot y x x y Length of a The length of any circular arc s is equal to the product of the measure of the radius Circular Arc of the circle r and the radian measure of the central angle that it subtends: s r If is the measure of the central angle expressed in radians and r is the measure of Area of a the radius of the circle, then the area of the sector, A, (region bounded by the 1 2 Circular Sector r central angle and the intercepted arc) is: A 2 If is the measure of the central angle expressed in radians and r is the measure of Area of a the radius of the circle, then the area of the segment, A, (region bounded by the 1 2 Circular Segment r sin intercepted arc and its chord)is: A 2 37 Precalculus Review- Basic Trig Test 1 Part 1 Name:_______________________________ Date:________________________________ Find the exact value of each expression: 1. sec 5. sin 150 8. Sketch a figure and find the coordinates for each circular point: 8 5 7 W W W b. c. 3 6 6 a. 3 2. tan 135 3. cot 270 6. cos 210 7. cos 315 4. sec d. 2 W 11 3 9. Without finding , find the exact value of csc if cos 7 and cot 8 0. 10. Without finding , find the exact value of sec if sin 8 and cot 9 0. 11. Find the values of the six trigonometric functions for angle the coordinates (5, -4) lies on its terminal side. in standard position if a point with 12. Find the values of the six trigonometric functions for angle coordinates (-3, -5) lies on its terminal side. in standard position if a point with the 13. If csc 15. Find the length of the missing side and the exact value of the six trigonometric functions of the angle in each figure: 2.5 , find sin a. 14. If cot 0.75 , find tan b. 7 11 8 7 38 Precalculus Review- Basic Trig Test 1 Part 2 Name:_______________________________ Date:________________________________ (1) (2) Convert from degrees to radians in simplest fractional form: a. 11.83 b. 47.5 c. 100 c. 7 rad 9 Convert from radians to degrees: a. 10 rad b. 1.3 rad (3) Find the measure of a central angle (4) Given a central angle of 18 , find the length of its intercepted arc in a circle of radius 5 feet. (Round to the nearest tenth.) (5) An arc is 1.5 feet long and is intercepted by a central angle of opposite an arc of 3 meters in a circle with a radius of 1 meter. 4 radians. What is the diameter of the circle? (Round to the nearest tenth.) (6) A sector has an arc length of 6 feet and a central angle of 1.2 radians. (a) Find the radius of the circle. (b) Find the area of the sector. (7) A sector has an area of 15 square inches and a central angle of 0.2 radians. (a) Find the radius of the circle to the nearest tenth. (b) Find the arc length of the sector to the nearest tenth. (8) Given a central angle of 20 , find the length of the radius of the circle, to the nearest tenth, whose intercepted arc has a length of 40 cm. (9) Steve rides his bike 3.5 kilometers. If the radius of the tire on his bike is 32 centimeters, determine the number of radians that a spot on the tire will travel during the trip. (10) Two gears are interconnected. The smaller gear has a radius of 3 inches, and the larger gear has a radius of 7 inches. The smaller gear rotates 250 . Through how many degrees, to the nearest tenth, does the larger gear rotate? (11) Using the accompanying diagram, find the area of the shaded region, to the nearest tenth, if a pentagon is inscribed in a circle that has a radius of 3.82 feet. (12) A regular octagon is inscribed in a circle with radius of 5 feet. Find the area of the octagon to the nearest tenth. 39 TRIGONOMETRY UNIT NOTES PART 2 40 Precalculus Lesson- Inverse Trig Functions Name:____________________________________ Date:_____________________________________ Objective: To learn to use inverse trig functions to solve for an angle or angles DO NOW: Find the exact value of sec 5 4 __________________________________________________________________________________________ What is an inverse trig function? What is it used for? Examples: 1. Write in the form of an inverse function: cos 2. Write in the form of an inverse function: cos 45 3. Solve by finding the value of x to the nearest degree: Sin 1 ( 1) x 4. Solve by finding the value of x to the nearest degree: Arc cos 1 2 x 2 2 Find each value (put angles in radian measure). Round any decimals to the nearest hundredth. 5. Arc tan 3 3 6. cos 2 Sin 1 3 2 41 Precalculus Lesson- Real world trig problems- SOHCAHTOA Name:_______________________________ Date:________________________________ Objective: solve real-world applications using right triangles and trigonometric functions (1) To measure the height of a cloud, you place a bright searchlight directly below the cloud and shine the beam straight up. From a point 100 feet away from the searchlight, the measure of the angle of elevation of the cloud is 83 12 . To the nearest tenth of a foot, find the height of the cloud. (2) A ranger spots a fire from a 73 foot tower in Polynomial Park. The measure of the angle of depression is 1 20 . To the nearest tenth of a foot, find how far the fire is from the tower. (3) A lighthouse 25 meters tall stands at the top of a vertical cliff. A boatman directly offshore finds that, to the nearest degree, the angles of elevation of the top and bottom of the lighthouse are 28 and 24 , respectively. To the nearest meter, find how far he is from the bottom of the cliff. 25 m 28 24 42 (4) A large, helium-filled penguin is moored at the beginning of a parade route awaiting the start of the parade. Two cables attached to the underside of the penguin makes angles of 48 and 40 with the ground and are in the same plane as a perpendicular line from the penguin to the ground. If the cables are attached to the ground 10 feet from each other, to the nearest tenth, how high above the ground is the penguin? 10 ft (5) A person standing 100 feet from the bottom of a cliff notices a tower on top of the cliff. The angle of elevation to the top of the cliff is 30 . The angle of elevation to the top of the tower is 58 . To the nearest tenth, find the height of the tower. 58 30 100 ft (6) Romeo stands 200 feet away from Juliet’s castle. He looks up to her balcony at an angle of elevation of 30 . He then sees a dove perched on the top of the castle, shifting his gaze to an angle of elevation of 40 . (a) If Romeo wants to climb up to Juliet’s balcony, how many feet, to the nearest tenth, would he have to climb? (b) To the nearest tenth, how many feet above the ground is the dove? 43 (7) While hiking on a level path toward Colorado’s front range, Brian determines that the angle of elevation to the top of Long’s Peak is 30 . Moving 1000 feet closer to the mountain, Brian determines the angle of elevation to be 35 . To the nearest foot, how much higher is the top of Long’s Peak than Brian’s elevation? (8) An exit ramp leading to a freeway overpass is 470 feet long and rises about 32 feet. What is the average angle of inclination of the ramp to the nearest tenth of a degree? (9) Tanya places her surveyor’s telescope on the top of a tripod 5 feet above the ground. She measures an 8 elevation above the horizontal to the top of a tree that is 120 feet away. To the nearest hundredth, find the height of the tree. 8 120 ft 5 ft (10) A 75 foot long conveyor is used at Function Farm to put hay bales up for winter storage. The conveyor is tilted to an angle of elevation of 22 . (a) Rounding to the nearest tenth, to what height can the hay be moved? (b) If Farmer Fred repositions the conveyor to an angle of 27 , how much higher can hay be moved compared to its initial configuration, to the nearest tenth? the 44 solve real-world applications using right triangles and trigonometric Junctions 0,„ ' [DO 1 71^ (4 Zf" A lighthouse 25 meters tall stands at the top of a vertical cliff A boatman directly offshore finds that, to the nearest degree, the angles of elevation of the top and bottom of the hghthouse are 28° and 24°, respectively. To the nearest meter, find how far he is from the bottom of the cliff. 77^ 2^° \ (3) X A ranger spots a fire from a 73 foot tower in Polynomial Park. The measure of the angle of depression is 1°20'.- To the nearest tenth of a foot, fmd how far the fire is ^^thetower. ^, . ^ (2) • Date: Name: To measure the height of a cloud, you place a bright searchlight directly below the cloud and shine the beam straight up. From a point 100 feet away fi-om the searchlight, the measure of the angle of elevation of the cloud is 83°12'. To the nearest tenth of a foot, find the height of the cloud. Objective: Precalculus Lesson- Real world trig problems ® (6) (5) (4) 1(^0. 7X0 m " - 2 = zoo /W."?' Jloo 6 _ (b) To the nearest tenth, how many feet above t (a) If Romeo wants to climb up to Juliet's balco tenth, would he have to climb? Romeo stands 200 feet away from Juliet's castl angle of elevation of 30°. He then sees a dove shifting his gaze to an angle of elevation of 40°. loo A person standing 100 feet from the bottom of a of elevation to the top of the cliff is 30°. The an the nearest tenth, find the height of the tower. Y-^lo A large, helium-filled penguin is moored at the parade. Two cables attached to the underside o ground and are in the same plane as a perpendic cables are attached to the ground 10 feet from e ground is the penguin? (7) While hiking on a level path toward Colorado's front range, Brian determine angle o f elevation to the top o f Long's Peak is 30° . Moving 1000 feet close mountain. Brian determines the angle o f elevation to be 35° . To the nearest much higher is the top o f Long's Peak than Brian's elevation? 3^ T l , 30-'~ X /c,^ -2.0 4- (8) A n exit ramp leading to a freeway overpass is 470 feet long and rises abou What is the average angle o f inclination o f the ramp to the nearest tenth o f 3Z (9) Tanya places her surveyor's telescope on the top o f a tripod 5 feet above an 8° elevation above the horizontal to the top o f a tree that is 120 feet a hundredth, fmd the height o f the tree. (10) A 75 foot long conveyor is used at Function Farm to put hay bales up for w storage. The conveyor is tilted to an angle o f elevation o f 22° . I (a) (b) Sir. Rounding to the nearest tenth, to what height can the hay be moved? I f Farmer Fred repositions the conveyor to an angle o f 27° . how much hi the hay be moved compared to its initial configuration, to the nearest te ?2 Q I - 3 ^ - 0 Precalculus Lesson: Law of Cosines Name:____________________________________ Date:_____________________________________ Objective: solve triangles by using the Law of Cosines Law of Cosines: 1. Suppose a triangle ABC has side a = 4, side b = 7, and angle C = 54º. What is the measure of side C? 2. Suppose a triangle XYZ has sides of x = 5, y = 6, and z = 7. What is the measure of the angle across from the side of measure 6? 45 3. Suppose a triangle ABC has side b = 2, side a = 5, and angle B = 27º. Find the measure of side c. 4. Suppose a triangle ABC has side b = 4, side a = 5, and angle B = 27º. Find the measure of side c. Exit Ticket: Complete on separate paper and hand in when finished. 1. 2. In a triangle PQR we have p = 8 and r = 11. Angle Q is 47º. What is the length of side q? A triangle XYZ has sides x = 1, y = 2, and z = 2.5. What is the measure of angle Y? 46 Precalculus Lesson- Forces and the Law of Cosines Name:____________________________________ Date:_____________________________________ Objective: To determine the resultant force vector when given two force vectors and an included angle. DO NOW: If m A 30 , AC=5, and AB=7, solve the triangle. Find all sides to the nearest tenth and angles to the nearest degree. __________________________________________________________________________________________ Force- push or pull upon an object resulting from the object's interaction with another object. Vector- a quantity of force having both magnitude and direction. Examples: 1. Two forces separated by 52 degrees acts on an object at rest. The magnitude of the two forces are 32 Newtons and 17 Newtons. Find the resultant force vector to the nearest Newton. 47 2. A game of “Three Way Tug-O-War” is being played by a group of students. Two of the students are trying to gang up on the other. They believe that it will be easier to win if they increase the angle they create with the third person. Is that true? Justify your answer by providing examples. 3. Two fisherman have hooked the same fish and they are trying to cooperatively reel it in. The angle the fisherman make with the fish is 87 degrees. If the first fisherman’s line has a maximum tensile strength 223 Newtons and the second fisherman’s line has a maximum tensile strength of 401 Newtons and the fishermans’ lines are at maximum strain, what is the resultant force applied to the fish? 4. What is the angle separating two component force vectors whose magnitude are 15N and 17N respectively if the resultant vector is 21N? 48 49 (2) (3) Given ABC where A = 13 , B = 65 20 , and a = 35: (a) Solve (i) (ii) (iii) (b) Find the area of ABC, to the nearest tenth, using the formula K ABC such that: C is in DMS form b is rounded to the nearest tenth c is rounded to the nearest tenth 1 bc sin A 2 Given GHJ where g = 45.7, H = 111.1 , and J = 27.3 : (a) Solve GHJ, rounding answers to the nearest tenth (b) Find the area of GHJ (to the nearest tenth) 50 Precalculus Lesson: Real world trig problems with Law of Cosines & Sines Objective: (1) Name:_______________________________ Date:________________________________ solve real-world applications using Law of Sines and Law of Cosines A derrick at the edge of a dock has an arm 25 meters long that makes a 122 angle with the floor of the dock. The arm is to be braced with a cable 40 meters long from the end of the arm back to the dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened? 40 m 25 m 122 (2) A triangular course for a 30 km yacht race has distances of 7 km, 9 km, and 14 km long. Find the largest angle of the course to the nearest degree. (3) A lamppost tilts toward the sun at a 2 angle from the vertical and casts a 25 foot shadow. The angle from the tip of the shadow to the top of the lamppost is 45 . Find the length of the lamppost to the nearest tenth of a foot. 2 45 25 ft 51 (4) Using the picture seen to the right, and rounding to the nearest tenth of a meter, find the height of the tree. 110 23 120 m (5) Using the picture seen to the right, and rounding to the nearest hundredth, find the area of the jib sail. 25 105 2.5 m (6) During an expedition, two hikers start at point A and head in a direction 30 west of south to point B. They hike 6 miles from point A to point B. From point B, they hike to point C and then from point C back to point A, which is 8 miles directly north of point C. To the nearest tenth of a mile, how many miles did they hike from point B B to point C? A 6 mi 30 8 mi C (7) The Shaffers plan to fence a triangular parcel of their land. One side of the property is 75 feet in length. It forms a 38 angle with another side of the property, which has not yet been measured. The remaining side is 95 feet in length. (a) Help the Shaffers by finding, to the nearest tenth, the length of fence needed to enclose this parcel of their land. (b) Using your answer from part (a) and the given information, find the area of this parcel to the nearest square foot. 52 (3) s \ (2) (boo ^ (1) C C o solve real-world applications using Law of Sines and Law of Cosines Date: Name:_ a'' + lol^ - So <x Qos 1 2 2 ° (22° or S'O Y- 13S' -25.° ! A lamppost tilts toward the sun at a 2° angle from tlie vertical and casts a 25 foot shadow. The angle from the tip of the shadow to the top of the lamppost is 45°. Find the length of the lamppost to the nearest tenth of a foot. C= A triangular course for a 30 km yacht race has distances of 7 km, 9 km. and 14 km long. Find the largest angle of the course to the neai-est degree. = C'-ZZ° A derrick at the edge of a dock has an arm 25 meters long that makes a 122° angle with the floor of the dock. The arm is to be braced with a cable 40 meters long from tlje end of the arm back to the dock. To the nearest tenth of a meter, how far from the edge of the dock will the cable be fastened? Objective: Precalculus Lesson: Real world trig problems with Law of Cosines & Sines (7) (6) (5) (4) ^ g3. .(3g ^COfO^-'i (oo _ 2 % U^- +• 1-5" (b) Using your answer from part (a) and tlie nearest square foot. (a) Help the Shaffers by finding, to the near enclose this parcel of their land. The Shaffers plan to fence a triangular parc feet in length. It foiTns a 3,8° angle vrith ano been measured. The remaining side is 95 f C"^- During an expedition, two hikers start at poi west of south to point B. They hike 6 miles point B, they hike to point C and then from miles directly north of point C. To the near did they hike from point B to point C? Using the picture seen to the right, and rou hundredth, fmd the area of the jib sail. Using the picture seen to the right, and rou of a meter, fmd the height of the tree. Precalculus Lesson: Name:____________________________________ Determining the number of Distinct triangles (ambiguous case) Date:_____________________________________ Objective: To determine the number of distinct triangles that can be formed given an angle and two consecutive sides DO NOW: The sides of a triangle measure 6, 7, and 9. What is the largest angle in the triangle? __________________________________________________________________________________________ Ambiguous Case: This is the case in the Law of Sines ( SSA) where there may be none, one, or two distinct triangles for which you can solve. There is a shortcut method to finding the number of distinct triangles that exist: Assume: Given two sides and one opposite angle: If a is acute: a b sin a no solution If a is obtuse: a b sin a one solution b a b sin a two solutions a b a b a b no solution one solution one solution Examples: How Many distinct triangles can be formed from the given information? 1. a 2 , b 3, m A 45 2. a 9, b 12, and m A 35 53 54 Trig Unit Part 2 Definitions, Laws & Formulas The following trigonometric identities hold for all values of expression is defined: Reciprocal Identities where each sin 1 csc cos 1 sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan The following trigonometric identities hold for all values of Quotient expression is defined: Identities sin cos tan sin cos where each cot Two angles in standard position that have the same terminal side. If is the degree Coterminal measure of an angle, then all angles measuring + 360k , where k is an integer, Angles are coterminal with . Since angles differing in degree measure by multiples of 360 are equivalent, every angle has infinitely many coterminal angles. A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. So, for any angle , 0 < < 360 , its reference angle is defined by: (a) , when the terminal side is in Quadrant I (b) 180 – , when the terminal side is in Quadrant II Reference Angles (c) – 180 , when the terminal side is in Quadrant III (d) 360 – , when the terminal side is in Quadrant IV If the measure of is greater than 360 or less than 0 , it can be associated with a coterminal angle of positive measure between 0 and 360 . This special relationship shows that cofunctions have equivalent values for complementary angles: sin = cos (90 – ) cos = sin (90 – ) Cofuntions tan = cot (90 – ) cot = tan (90 – ) sec = csc (90 – ) csc = sec (90 – ) Degree & Radian 1 radian Conversion Formulas 1 deg ree 180 deg rees (or about 57.3 ) 180 radians (or about 0.017 radian) Coordinates of for any points (x, y) on the unit circle, given angle , W( ) = (cos , sin ) Circular Points 55 For any angle in standard position with measure , a point P(x, y) on its terminal Trigonometric Functions of an Angle in Standard Position x2 side, and r sin csc Inverses of the Trigonometric Functions y 2 , the trigonometric functions of are as follows: y r r y cos x r tan sec r x cot y x x y Trigonometric Function y = sin x Inverse Trigonometric Relation x = sin-1 y or x = arcsin y y = cos x x = cos-1 y or x = arccos y y = tan x x = tan-1 y or x = arctan y Length of a The length of any circular arc s is equal to the product of the measure of the radius Circular Arc of the circle r and the radian measure of the central angle that it subtends: s r If is the measure of the central angle expressed in radians and r is the measure of Area of a the radius of the circle, then the area of the sector, A, (region bounded by the 1 2 Circular Sector r central angle and the intercepted arc) is: A 2 If is the measure of the central angle expressed in radians and r is the measure of Area of a the radius of the circle, then the area of the segment, A, (region bounded by the 1 2 Circular Segment r sin intercepted arc and its chord)is: A 2 Let ABC be any triangle with a, b, and c, representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the Law of Sines a b c following is true: sin A sin B sin C Let ABC be any triangle with a, b, and c, representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the a2 = b2 + c2 – 2bc cos A Law of Cosines following are true: b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C Let ABC be any triangle with a, b, and c, representing the measures of the sides opposite the angles with measurements A, B, and C, respectively. Then the area K can be determined using one of the following six formulas: Area of Triangles K K 1 bc sin A 2 1 2 sin B sin C a sin A 2 K K 1 ac sin B 2 1 2 sin A sin C b sin B 2 K K 1 ab sin C 2 1 2 sin A sin B c sin C 2 56 Precalculus Review- Trig Test 2 Name:_______________________________ Date:________________________________ ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK! (1) A boatman is heading for the entrance of Hypotenuse Harbor in Trigonometry Town. At one point, he measures the angle of elevation to the base of a lighthouse to be 10 . After traveling for 5 miles, the angle of elevation measures 30 . (a) Find, to the nearest tenth, how far away the boatman is from the entrance to the harbor when he took the second angle reading. (b) Using your answer from part (a), find, to the nearest tenth, how far away the boatman is from the entrance to the harbor when he took the first angle reading. (c) Using your answer from part (a), find, to the nearest tenth, how high the base of the lighthouse is above the entrance to the harbor. Hypotenuse Harbor 5 mi (2) The rim of a basketball hoop is 10 feet above the ground. The free-throw line is 15 feet from the basket rim. If the eyes of a basketball player are 6 feet above the ground, what is the angle of elevation, to the nearest tenth of a degree, of the player’s line of sight when shooting a free throw to the rim of the basket? (3) Looking out across Hypotenuse Harbor is a lighthouse that stands 175 feet tall. How far from shore, to the nearest tenth, is a sailboat if the angle of depression from the top of the lighthouse is 13 15 ? (4) A soccer game is being played in Polynomial Park. Steve is standing 35 feet from one post of the goal and 40 feet from the other post. Howie is standing 30 feet from one post of the same goal and 20 feet from the other post. If the goal is 24 feet wide, which player has a greater angle to make a shot on goal? Show an algebraic solution and explain your answer in detail. (5) Two adjacent apartment buildings in Geometry Garden Estates share a triangular courtyard. They plan to install a new gate to close the courtyard that forms an angle of 10 48 with one building and an angle of 48 20 with the second building, whose length is 527 feet. (a) Find, to the nearest tenth, the area of the courtyard. (b) Find, to the nearest tenth, the length of this new gate. (6) Anthony, Bill, and Chris all live in Trigonometry Town. Anthony lives on Angle Avenue and is 6.4 miles away from Chris’ house. Bill lives on Binomial Boulevard and is 3.8 miles away from Chris’ house. Chris lives on the corner of Angle Avenue and Binomial Boulevard, where the two streets intersect and form an angle that measures 67 40 . To the nearest tenth of a mile, find the distance between Anthony’s house and Bill’s house. 57 ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK! (7) Steve is competing in Trigonometry Town’s annual canoe race, which is to be run over a triangular course marked by buoys A, B, and C. The distance between A and B is 100 yards, between B and C is 160 yards, and between C and A is 220 yards. In DMS form, and rounded to the nearest minute, find the measure of the largest angle within this course. (8) A landscaper wants to plant begonias along the edges of a triangular plot of land in Polynomial Park. Two angles of the triangle measure 95 and 40 . The side between these two angles is 80 feet long. (a) Find, to the nearest tenth, the missing sides of this triangular plot of land. (b) Using your answers from part (a), find, to the nearest tenth, the perimeter of this triangular plot of land. (c) Using your answers from part (a), find, to the nearest tenth, the area of this triangular plot of land. (9) Lafawnduh and Towanda are flying kites on a windy spring day. Lafawnduh has released 250 feet of string, and Towanda has released 225 feet of string. The angle that Lafawnduh’s kite string makes with the horizontal is 35 , while the angle that Towanda’s kite string makes with the horizontal is 42 . (a) Draw a labeled diagram to model this situation. (b) Which kite is higher and by how much (to the nearest tenth)? (10) A landscaper is developing the grounds of Polynomial Park. He has sketched plans for the area, as shown in the quadrilateral PQRS seen to the right. (a) A walkway is to be constructed from point S to Q. Find, to the nearest tenth of a foot, the length of this walkway. (b) Using your answer from part (a), find, to the nearest tenth, the area of the park. (11) Find all solutions for each ABC given the following information, using the Laws of Sines and Cosines. If no solutions exist, write none. Round each answer to the nearest tenth. (a) a = 32.9 ft, b = 42.4 ft, c = 20.4 ft (b) c = 15 in, A = 42 , B = 68 (c) a = 172 mi, c = 203 mi, A = 38.7 (d) A = 92.6 , B = 88.9 , a = 15.2 cm 58 ® s (4 r, 3:1.• ^0.^ V-Z-- Y V 2o • - a (I'z -^¥20. w jeer A o c = i/^.b 73-7 b or /3i?.'/ ^-^