Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Highlights for Trigonometry Final Exam No Calculator: Finding the exact values of the six trig functions (#1, 5, 6, 7, 25) Converting from degrees to radians, and from radians to degrees (#7) Possible/impossible values for the six trig functions (domain and range) (#2, 3) Graph and list properties of the trig functions, especially sine and cosine (#9, 10) f ( x) c a sin b( x d ) Gateway problems (#13-15) Solving trig equations (#16) Verifying basic trig identities (must know fundamental identities) (#4) Sum and difference/double and half angle applications (#11, 12) Conversions between polar and rectangular forms (also in complex number system) (#21-24) Calculator: Solve oblique triangles (#16-18) Find the area of a triangle (#19) Evaluate any one of the six trig functions (including DMS notation) (#5-7, 12-15) Find arc length and/or sector area (#10) Solve right triangle problems (#3, 4, 8, 9) 1. Sketch the angle in standard position and find all six trigonometric functions for the point (−12, 5). 2. In which quadrant is each of these true? a. sin 𝜃 < 0 and tan 𝜃 > 0 b. cos 𝜃 < 0 and csc 𝜃 > 0 c. cot 𝜃 < 0 and sec 𝜃 > 0 3. Possible or not possible? (Circle one.) a) sec 𝜃 = − 3 2 POSSIBLE NOT POSSIBLE b) tan 𝜃 = −4 POSSIBLE NOT POSSIBLE c) sin 𝜃 = 0.5 POSSIBLE NOT POSSIBLE POSSIBLE NOT POSSIBLE d) cos 𝜃 = 4 3 4. Solve cot (5𝜃 + 2°) = tan(2𝜃 + 4°) for θ. 5. Find the exact values of all six trig. functions for an angle of 480°. 6. Find the exact value of the trigonometric function,. a) cos 225° b) tan(−210°) c) csc 330° 7. Convert from degrees to radians (or radians to degrees.) a) 225° b) −315° c) 7𝜋 6 d) − 8. Find the exact values: a) sec 4𝜋 3 b) sin(− 5𝜋 ) 6 𝜋 c) tan 2 4𝜋 3 9. Fill in the following chart. Equation Amplitude Period Phase Shift 1 a. 𝑦 = 2 sin 𝑥 1 2 b. 𝑦 = cot 3𝑥 c. 𝑦 = −3 sin 2𝑥 − 1 d. 𝑦 = − sin (𝑥 − 3𝜋 ) 4 𝜋 e. 𝑦 = cos (3𝑥 + 2 ) f. 1 3 𝜋 4 𝑦 = 2 + 4 tan (𝑥 − ) 10. Sketch two periods of the graph of each of the following functions. a) 𝑦 = tan 𝑥 b) 𝑦 = csc 𝑥 𝜋 c) 𝑦 = −2 cos (𝑥 − 2 ) d) 𝑦 = 3 sin 2𝑥 − 1 11. Vertical Shift Let sin 3 8 and cos , with both θ and β in QIV. Find the exact value for 5 17 each. a. cos 𝜃 b. sin c. cos(𝜃 − 𝛽) d. sin(𝜃 + 𝛽) e. sin 2 f. tan 2 12. Find the exact value of cos 75°. 13. Give the exact value of y (in radians). 1 𝑎) 𝑦 = sin−1 ( ) √2 𝑐) 𝑦 = arccot(−1) 𝑏) 𝑦 = arccos (− 𝑑) 𝑦 = csc −1(1) 14. Give the exact value of y (in degrees). 𝑎) 𝑦 = tan−1 0 𝑏) 𝑦 = arcsec(2) √3 ) 2 2 3 15. Evaluate tan (sin−1 (− )). 16. Solve the following trigonometric equations. a) tan2 𝑥 − 1 = 0 17. (6 − 5𝑖) + (2 + 7𝑖) 19. (3−𝑖) (2+5𝑖) 21. Write the complex number in polar form : b) 2 sin2 𝑥 − 3 sin 𝑥 − 2 = 0 18. (2 + 2𝑖)(4 − 3𝑖) 20. Solve 2𝑥 2 + 3𝑥 = −4 3 − 3𝑖√3 22. Write the complex number in rectangular form: 2cis225° 23. Multiply: 5cis20° and 2cis130° and write your answer in rectangular form. 24. Divide: 6(cos 30° + 𝑖 sin 30°) and 2(cos 60° + 𝑖 sin 60°). Write your answer in rectangular form. 25. Fill in the following table: 𝜃=0 sin cos tan cot sec csc 2 𝜃= 3𝜋 2 1. Find the measures of the two angles. (Don’t just find 𝑥.) 7𝑥 + 3 10𝑥 + 7 2. Evaluate and leave answer in requested form: a. 90° − 36° 18′ 47" (DMS) b. 124° 12′ 55" − 230° 35′ 16" (Decimal degrees) 3. If two angles of a triangle are 136° 50′ and 41° 38′, find the third angle. 4. Refer to the picture at right. Find the value of 𝑥. Find a decimal approximation correct to 4 decimal places. 5. cot(−512°20′ ) 6. sec(58.9041°) 7. sin(243°12′ ) Solve the right triangle. 68.5142° B 8. 3579.42 m a C A b 9. The angle of depression of a television tower to a point on the ground 36 meters from the bottom of the tower is 29.5°. Find the height of the tower. 10. Suppose that a windshield wiper is 10 inches long and rotates back and forth through an angle of 95°. What is the area of the region cleaned? 11. Find the linear speed of a point on the edge of a flywheel of radius 7 cm if the flywheel is rotating 90 times per second. Evaluate and round to the nearest hundredth. 12. 𝜃 = arctan(1.78) 13. 14. 𝜃 = cot −1(4.505) 15. 𝜃 = arcsec(3.4723) 𝜃 = sin−1 (−.66) 16. Solve the triangle given that 𝐴 = 68.41°, 𝐵 = 54.23°, 𝑎 = 12.75 𝑓𝑡. 17. Solve the triangle given that 𝐴 = 38.5°, 𝑎 = 9.72 𝑘𝑚, 𝑏 = 11.8 𝑘𝑚 18. Solve the triangle given that 𝐴 = 41.4°, 𝑏 = 2.78 𝑦𝑑. , 𝑐 = 3.92 𝑦𝑑. 19. Find the area of the triangle with 𝐴 = 35°, 𝑏 = 5 𝑓𝑡, 3 𝑓𝑡.