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UNIT 7: TRIGONOMETRY BASICS CW #1: Right Triangle Trig and Special Right Triangles Find the measure of x in each of the following. Leave irrational answers in radical form: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet long, what is the measure of the angle that the ramp makes with the ground? 14. From a point on the ground that is 100 feet from the base of a building, the tangent of the 5 angle of elevation of the top of the building is . To the nearest foot, how tall is the 4 building? 15. Find the measure of the diagonal of a rectangle whose sides measure 14 and 48. 16. The length of a rectangle is 7 meters more than its width. A diagonal of the rectangle measures 17 meters. Use algebra (not guess and check!) to find the length of the rectangle. 17. In a triangle, the lengths of the sides are 7, 8, and 12. Is this a right triangle? Show your reasoning. CW# 2: Creating a unit circle “calculator”: Coordinate Geometry, Rotations, Unit Circle, Degree Measure, more Vocabulary DEFINE Unit Circle: I. Determine the quadrant in which an angle of the given measure lies: _______________ 1. 140 ________________ 2. 210 _______________ 3. 97 ________________ 4. 315 _______________ 5. 80 ________________ 6. 240 _______________ 7. 168 ________________ 8. 200 II. Find the angle of smallest positive measure coterminal with an angle of the given measure: _______________ 9. 400 ________________ 10. 710 _______________ 11. 580 ________________ 12. 790 _______________ 13. 30 ________________ 14. 370 _______________ 15. 400 ________________ 16. 800 III. Draw a sketch of an angle of the given measure, indicating the direction of the rotation by an arrow. 17. 18. 100 210 19. 140 IV. Determine the quadrant in which an angle of the given measure lies: _______________ 1. 475 ________________ 2. 428 CW #3: Creating a unit circle “calculator”: Radian Measure & Conversions 1. Define Unit Circle: 2. Define RADIAN: _______________3. One radian is approximately how many degrees? _______________4. How many radians are in the circumference of a circle? _______________5. How many radians are in ½ the circumference of a circle? _______________6. How many radians are in ¼ the circumference of a circle? _______________7. What are the coordinates of the point at 0 and 360 degrees on the unit circle? _______________8. What are the coordinates of the point at 90 degrees on the unit circle? _______________9. What are the coordinates of the point at 180 degrees on the unit circle? _______________10. What are the coordinates of the point at 270 degrees on the unit circle? 11. is the measure of a central angle that intercepts an arc of length s in a circle whose radius has length r . _______________ a. If s 12 and r 4, find . _______________ c. If r 4 and 2.5, find s . _______________ b. If s 12 and 6, find r . 12. On a clock, the length of the pendulum is 30 centimeters. A swing of the pendulum determines an angle of 0.8 radian. Find, in centimeters, the distance traveled by the tip of the pendulum during this swing. 13. Convert the following to radian measure: _______________ a. 180 _______________b. 90 _______________ d. 360 _______________e. 60 _______________ c. 45 _______________ f. 30 _______________ g. 270 14. _______________h. 150 _______________ i. 240 Convert the following to degree measure: ___________ a. 1 radian ____________b. 2 radians _______________ d. _______________e. _______________ g. _______________ j. 10 3 4 _______________h. _______________k. 3 2 7 2 ___________ c. 5 radians _______________ f. 9 _______________ i. 5 6 _______________ l. 4 3 CW #4: Creating a unit circle “calculator”: Unit Circle Trigonometry 1. Define UNIT CIRCLE: 2. Define RADIAN: 3. What are the COORDINATES of the point at 90 degrees on the unit circle? 4. What are the COORDINATES of the point at 30 degrees on the unit circle? 5. What are the COORDINATES of the point at 45 degrees on the unit circle? 6. Give the EXACT value of each of the following (NO DECIMALS … NO CALCULATORS!) a. sin 90 b. cos 45 c. tan 90 d. cos 30 e. sin 60 f. cos180 g. tan 270 h. tan 30 i. sin j. sin k. cos m. sin p. sin 4 3 4 n. cos q. tan 2 3 11 6 6 l. tan o. cos 3 2 r. cos 4 3 7. What is the SLOPE of every vertical line? 8. What is the SLOPE of every horizontal line? 9. Write the formula for tangent in terms of sine and cosine: 10. Write the formula for tangent in terms of x and y: 11. Use the idea that tangent = slope to find the EXACT value of each of the following: a. tan d. tan g. tan 0 6 11 6 b. tan e. tan h. tan 315 4 3 c. tan f. tan i. tan 3 4 2 2 3 CW #5: Creating a unit circle “calculator”: Mastering Unit Circle Trigonometry 1. Fill in the following unit circle completely and accurately. (Give each point all of its’ names) Find the exact value of each of the following: __________ 1. sin 45 __________ 2. cos __________ 3. tan 3 2 __________ 4. sin __________ 5. cos __________ 6. tan __________ 7. tan __________ 8. cos __________ 9. sin __________ 11. tan 2 6 3 4 __________ 13. sin 2 __________ 15. cos 3 6 4 __________ 10. cos __________ 12. sin __________ 14. tan 2 3 6 Find the EXACT value of each expression: 16. tan(315 ) tan(135 ) 17. sin(300 ) sin(240 ) CW #6: Reference Angles MUST DO: Express the given function as a function of a positive acute angle. 1. sin100 2. cos190 3. sin 248 4. sin 98 5. tan 345 6. tan 620 7. sin(20 ) 8. cos 290 9. sin(158 ) 10. tan 300 11. cos(200 ) 12. tan(80 ) 13. sin 340 14. cos150 15. tan 237 CW #7: Reciprocal Trig Functions Find the EXACT numerical value of each expression: 1. csc150 2. sec 240 3. cot 315 4. csc120 5. sec2 6. cot 3 2 8. sec 10. csc 12. cot 2 7. csc 9. sin tan 3 11. cos 2 4 6 sec 2 4 6 5 4 2 sec 3 csc 2 Write each expression in terms of sin , cos , or both. Simplify whenever possible. 1. tan 2. cot 3. sec 4. csc 5. tan csc 6. tan sec cot csc 8. sin csc 9. sec csc 7. Name the quadrant in which angle B must lie: 10. cot B 0 and sin B 0 11. sec B 0 and tan B 0 CW #8: Parent Trig Functions MUST DO: Accurately sketch each of the following parent trig functions in the interval 0 2 without a calculator: 1. f ( x) sin x 2. f ( x) cos x 3. f ( x) tan x Use your knowledge of transformation of parent functions (unit 1) to graph each of the following without a calculator.: 4. f ( x) sin x 2 5. f ( x) cos x Use your knowledge of transformation of parent functions (unit 1) to graph each of the following without a calculator: 6. f ( x) sin( x 2 ) 7. f ( x) 2 cos x CW #9: Amplitude, Frequency, Period, Phase Shift & Midline of Trig Graphs I. State the amplitude, frequency, period, midline, and phase shift (if applicable) of each of the following functions: 1. 3. 5. f ( x) 2 cos x 2. f ( x) cos 2 x Amplitude = __________ Amplitude = __________ Frequency = __________ Frequency = __________ Period = _____________ Period = _____________ Midline = _______________ Midline = _______________ Phase Shift = _______________ Phase Shift = _______________ 1 f ( x) sin 3x 2 4. f ( x) 2sin 4 x 3 Amplitude = __________ Amplitude = __________ Frequency = __________ Frequency = __________ Period = _____________ Period = _____________ Midline = _______________ Midline = _______________ Phase Shift = _______________ Phase Shift = _______________ 1 f ( x) sin x 2 2 6. f ( x) sin x 1 Amplitude = __________ Amplitude = __________ Frequency = __________ Frequency = __________ Period = _____________ Period = _____________ Midline = _______________ Midline = _______________ Phase Shift = _______________ Phase Shift = _______________ For #7-13, graph the function in the interval x 2 without a calculator. 7. f ( x) 2sin x 8. f ( x) cos 2 x 9. 1 f ( x) 2 cos x 2 10. f ( x) sin 2 x 2 11. f ( x) sin x 4 12. f ( x) 3cos( x ) 13. f ( x) 5cos 2 x 1 2 (Change the scale on the y-axis to make it fit) CW #10: Modeling with Trigonometry In #1-4, write a function for the sinusoid. 1. 2. 3. 4. 5. The lowest frequency of sounds that can be heard by humans is 20 hertz. The maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write a sine model that gives the pressure P as a function of time t (in seconds). 6. The table shows the numbers of employees N (in thousands) at a sporting goods company each year for 11 years. The time t is measured in years, with t=1 representing the first year. (a) Use sinusoidal regression to find a model that gives N as a function of T. (b) Use your model to predict the number of employees at the company in the 12th year. t N 1 20.8 2 22.7 3 24.6 4 23.2 5 20 6 17.5 7 16.7 8 17.8 9 21 10 22 11 24.1 CW #11: Using Trig Identities 1. An identity is a statement that is always ________________. In #2-5, Find the values of the other five trigonometric functions of . 2. 5 3 csc , 3 2 3. 1 sin , 0 3 2 4. 3 tan , 7 2 4. 5 3 cos , 6 2 In #6-9, simplify the expression 6. sin x cot x 7. cos 1 tan 2 8. cos 2 cot 2 9. sin sec 2 In #10-13, verify each identity. 10. 12. sin x csc x 1 cos 1 2 1 1 sin( ) 11. tan csc cos 1 13. 1 cos x sin x 2 csc x sin x 1 cos x CW #12: Using Sum & Difference Formulas 1. Write the expression cos130 cos 40 sin130 sin 40 as the cosine of an angle. In #2-7, find the exact value of the expression. 2. tan 15 4. sin 165 6. 11 cos 12 3. tan 195 5. cos 105 7. 17 tan 12 In #8-11, simplify the expression. 8. tan x 9.