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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)
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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.