Download final-sample

MA140 Sample Final Name:
20-point problems. Show all your work. Circle your answers. Sufficient work
must be shown to receive any credit.
1. If tan x = − 43 with 0 < x < π, find the exact value of tan (x + π).
2. If tan x = − 43 with 0 < x < π, find the exact value of cot x.
3. If tan x = − 43 with 0 < x < π, find the exact value of sin x.
4. Given that A = 25° , a = 15, and b = 33, find the measure of angle B to the nearest degree. If
there are two answers, give both of them. If there are no possible answers, write “none”.
1
5. Suppose sin x =
1
3
with
π
2
< x < π. Find the exact value of sin 2x.
6. Find all exact solutions of the equation
√
2 sec x + 2 = 0 on the interval [0, 2π].
7. Suppose that a supporting cable runs from the top of a 25 foot antenna to a point on the ground
20 feet from the base of the antenna. What is the angle between the cable and the ground (to the
nearest degree)?
2
8. Referring to the diagram below, find the exact value of sin θ.
9. Referring to the diagram in problem #8, find the exact length of arc s cut off by the angle θ.
5
10. Suppose sin x = − 13
with
sin(x + y).
3π
2
< x < 2π and cos y = − 53 with π < y <
3π
.
2
Find the exact value of
11. Given that B = 110° , C = 39° , and b = 42 cm, find a to the nearest centimeter. If there are two
answers, give both of them. If there are no possible answers, write “none”.
3
csc x − sin x
=
cos x
(Hint: it is one of the six trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x)
12. Fill in the blank:
30-point problems. Show all work. Circle your answers.
13. Suppose tan x = − 34 with π < x < 2π. Find the exact value of cos x2 .
14. To find the length AB of a small lake, a surveyor at point C measures angle ACB to be 115°, length
AC to be 500 feet, and length BC to be 325 feet. What is the length of the lake (to the nearest
foot)? Circle your answer.
4
15. Find the amplitude, period, phase shift, and vertical shift of the function y = sin x3 − π3 . Also,
graph the function over an interval of at least one period in length. Be sure to provide sufficient
labels on your graph.
Amplitude:
Period:
Phase Shift:
Vertical Shift:
16. Verify the identity given below, using algebraic manipulation and trigonometric identities. Be sure to
show all steps of your solution. Graphical solutions are not allowed for this problem.
1 + cos x
sin x
+
= 2 csc x
1 + cos x
sin x
5
17. Find all exact solutions for θ in degrees, 0° ≤ θ < 360°, of the equation:
2 cos2 θ = 1 − sin θ
18. If a force is applied using a rope with 100 lb tension at 24° above horizontal, find the horizontal and
vertical components of the force (to the nearest pound.)
Vertical:
Horizontal:
19. Derive the sum of angles formulas for sine and cosine using Euler’s identity.
6