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MA 10A Arcadia Valley Career Technology Center Topic: Basic Trigonometry Show-Me Standards: MA2, MA4 Tech Math I Credit Last Update: March 2008 Focus: Solving Right Triangles ( Six Trig. Functions,) MO Grade Level Expectations: G1A9, G1A11, G2A8, G4B10 NCTM Standards: 8D, 10A OBJECTIVE: The students will be able to find the values of the six basic trigonometric functions and solve problems involving right triangles. *Recall from Lesson 7B RIGHT ANGLES: angles that measure 90-degrees. A right angle is often shown with a small square drawn in the corner of the angle. ¬ RIGHT TRIANGLE: c=hypotenuse – by definition, the side opposite the right angle. a=leg b=leg c 2 a 2 b 2 , or Pythagorean Theorem: a c2 b2 b c2 a2 , where c is the hypotenuse. c a2 b2 OTHER IMPORTANT INFORMATION: 1. The sum of the angles of any triangle is 180-degrees. 2. The sum of the two acute angles of the right triangle is 90-degrees. 3. A triangle without a right angle (an ‘oblique triangle’) can be worked with by first creating two right triangles. Working from the known values, the two triangles can be solved and results combined to give the desired angles and sides of the oblique triangle. opposite hypotenuse adjacent 4. The Trigonometric Functions are: cos , another way of remembering this hypotenuse opposite tan adjacent Old Some Horse Another information is as follows: Caught . Horse Oats Taking Away sin Now, we are going to look at the other three basic Trigonometric Functions: cosecant (csc) secant(sec) cotangent(cot) csc θ = hypotenuse opposite sec θ = hypotenuse adjacent cot θ = adjacent opposite - Notice that the sin, cos, and tan are reciprocals of the csc , sec, and cot respectively. Therefore, the following are true: csc θ = _ 1__ sin θ sec θ = 1__ _ cos θ cot θ = 1 ___ tan θ Examples: If sin θ = 4_ or 0.8 , then find the other five trig. function values. 5 - we can use the Pythagorean Thm. to find the missing leg. The opposite leg must be 4 and the hypotenuse is 5 so, (adj.leg)2 + 42 = 52 and adj. leg = 3 - cos θ = 3 _ 5 or 0.6 tan θ = 4_ 3 or 1.3333 csc θ = 5_ 4 or 1.25 sec θ = 5_ 3 or 1.6667 cot θ = 3_ 4 or 0.75 *Recall from Lesson 7B - Inverses of the basic trig. functions are used when you know the value of the trig. function but you would like to know the measure of the angle that goes with it. (symbol for inverse is -1 ) Examples: sin-1 (½) = 30° because the sin 30° = ½ cos-1 (.5667) = 56.2° Solving a Right Triangle – if you know the measures of any two sides of a right triangle or the measure of any one side and one of the acute angles, you can find all the missing measures. This is called solving the right triangle. A - Examples: c b C a ABC is a right triangle B Ex 1. If a = 10 “ and Angle B = 35°, then solve the right triangle. Step 1 tan 35° = b_ , 0.7002 = 10 b_ , 10 b = 7.0” Step 2 90° - 35° = 55° , Angle A = 55° Step 3 cos 35° = 10_ c or , 0.8192 = 102 + 7.02 = c2 , √ 149 = c 10_ c , , c = 12.2” c = 12.2” So, a = 10” b = 7.0” c = 12.2” Angle A = 55° Angle B = 35° Angle C = 90° Ex 2. If b = 6 cm and c = 13 cm , then solve the right triangle. Step 1 a2 + 62 = 132 , a2 + 36 = 169 , a = √133 , a = 11.5 cm Step 2 sin-1 6_ = Angle B , 13 Angle B = 27.5° Step 3 90° - 27.5° = 62.5° , Angle A = 62.5° So, a = 11.5 cm b = 6 cm c = 13 cm Angle A = 62.5° Angle B = 27.5° Angle C = 90° Note: There is usually more than one way to solve a right triangle. You may want to go back and redo the two examples using different steps. The answers should stay the same. Problems: 1. Find the values for the five other trigonometric functions given sin θ = 8_ . 17 2. Find the values for the five other trigonometric functions given sec θ = 16_ 12 3. Solve the right triangle given the following measures: Angle A = 35° , a = 12 mm 4. Solve the right triangle given the following measures: a = 4 ft. , b = 7 ft. 5. Solve the right triangle given the following measures: Angle A = 17° , c = 3.2 miles . SOLVE: 6. A sightseeing boat is near the base of a waterfall and the captain estimates the angle of elevation to the top of the falls to be 30°. If the falls are 180 ft. high, how far is the boat from the base of the falls? 7. A surveyor stands 200 ft. from a bridge pillar. The bridge pillar is 320 ft. high. What is the surveyor’s angle of sight to the top of the pillar? 8. A tread mill is 48” long and set on a 6° incline. What is the vertical rise of the treadmill? 9. The tailgate of a moving truck sits 3’ above the ground with a ramp out the back at a 15° incline with the ground. How long is the ramp? 10. A 38’ guide wire runs from the top of a pole to the ground. The wire makes an angle with the ground of 52°. How far is the pole from the anchored guide wire? How tall is the pole?