Download Final Exam Review

Math 104 Final Exam Review
1.
Find all six trigonometric functions of θ if (−3, 7) is on the terminal side of θ .
2.
Find cos θ and sin θ if the terminal side of θ lies along the line y = −2 x in
quadrant IV.
3.
Find the remaining trigonometric functions of θ if cscθ = 3 and θ terminates
in quadrant II.
4.
Use identity substitutions to simplify: a) cscθ − cot θ cos θ
5.
Show that cos θ (secθ + tan θ ) =
1 + sin θ by transforming the left side into the right side.
6.
Simplify the expression
7.
If sin θ =
8.
In which quadrant will θ lie if cscθ > 0 and cos θ < 0 ?
9.
In ∆ABC , C = 90° , c= 4.79 cm, and b = 3.68 cm. Draw the triangle then find each of the
following:
a) Side a
b) Angle A
c) Angle B
10.
In ∆ABC C = 90° , A = 60 , and side a = 12 cm. Find exact answers for each of the
following.
a) Side c
b) Side b
11.
Use a calculator to find: a) tan 6350′
your answers to two decimal places)
12.
A CB antenna is located on the top of a garage that is 16 feet tall. From a point on level
ground that is 100 feet from a point directly below the antenna, the antenna subtends an
angle of 12 . Approximate the length of the antenna to one decimal place.
b) (1 − cos θ )(1 + cos θ )
9 − x 2 as much as possible after substituting 3sin θ for x.
1
with θ in quadrant I, find cos θ , cscθ , and cot θ .
a
12
100′
b) θ if θ is acute and secθ = 1.923 (Round
16′
13.
A pilot, flying at an altitude of 5000 feet, wishes to approach a landing point on a runway
at an angle of 10 (angle of depression). Approximate, to the nearest 100 feet, the distance
from the airplane to the landing point at the beginning of the descent.
14.
Give the exact values of: a) sin
15.
Find the exact values of: a) sec 45°
16.
Show that cotangent is an odd function.
17.
Convert to radians: a) −120
18.
Convert to degrees: a)
19.
Draw the following angles in standard position and find the reference angle:
13π
a) 272 12′ b) −120
c)
12
20.
Use a calculator to find θ if θ is between 0 and 360 and
4π
3
2π
3
 3π 
b) 4 cos  − 
 4 
b) csc 60°
c) csc
5π
6
c) sin 2 60° + cos 2 45° (Simplify)
b) 250
b)
7π
12
a) cos θ = −.4772 with θ in quadrant III (Round to the nearest tenth.)
b) secθ = 1.545 with θ in quadrant IV (Round to the nearest tenth.)
21.
If θ =
π
2π
is a central angle that subtends an arc length of s = , find the radius of the
3
4
circle.
22.
Find the area of the sector formed by central angle θ = 2.4 in a circle of radius r = 3 cm.
23.
A conical paper cup is constructed by removing a sector from a circle of radius 5 inches
and attaching edge OA to OB (see figure). Find angle AOB so that the cup has a depth of
4 inches.
B
O
A B
A
O
4′′
24.
Find the amplitude, period, and phase shift of:
a.
=
y
b.
25.
π
π
3 cos  x − 
2
4
π
1
y=
−5sin  x + 
6
3
Sketch one period of:
a.
1
π
y = − sin 
2 2

x

π
1
b.=
y 3cos  x + 
2
6
26.
c.
y = −6 cot ( 3 x )
d.
1 
y = 6 csc  x 
3 
Find the equations of each of the following:
a.
b.
c.
27.
π

Find an interval over which the graph
of y cos  π x +  completes one complete cycle.
=
2

28.
1
Find the range of
=
y 3cos x + (3 + 2) .
4
29.
Find the exact value for each of the following:
a.
30.
1

sin  sin −1 
2

b.
 π
sin −1  sin 
4

c.
Find the exact values for each of the following:
a.
π 

cos −1  cos(π − ) 
19 

b.
π 

cos −1  cos(π + ) 
19 

2π 

sin −1  sin

3 

31.
Find the exact value for each of the following:
a.
3π 

sin −1  tan

4 

b.
  π 
cos −1  sin  −  
  6 
c.

 2 
sin  arccos  −  
 3 

d.
1
4

sin  arctan − arccos 
2
5

32.
If −1 ≤ x ≤ 1 , rewrite cos(sin −1 x) is terms of x without trig functions.
33.
Let sin θ =
34.
a. Derive the formula for tan( A − B) using the formulas for sine and cosine.
4
12
with θ in the second quadrant and sin α = −
with α in the third quadrant.
5
13
Compute sin(θ + α ) .
1
b. Find tan A if tan( A − B) =
9 and tan B = .
3
35.
If x is a positive number, find:
a.
36.
b.
tan(cos −1 3 x)
Find the exact values of:
a.
37.
cos(2sin −1 x)
sin 22.5
Let sin A = −
a.
b.
tan
π
c.
12
cot15
3
12
with A in quadrant IV and sin B =
with B in quadrant II and find:
5
13
sin( A + B )
b.
cos 2B
c.
sin
A
2
38.
Prove the following identities:
cos x
1 + sin x
=
1 − sin x
cos x
b. =
sec x sin x(tan x + cot x)
sec x − cos x =
sin x tan x
c.
sin 2 x
d.
cot x =
1 − cos 2 x
a.
π
π
39.
If x = a sin θ , −
40.
Express as a single trig function and then simplify:
sin
π
12
cos
2
≤θ ≤
2
and a > 0 , express
a 2 − x 2 in terms of a trig function of θ .
π
5π
5π
+ cos sin
12
12
12
41.
Solve 2sin θ − 3 =
0 for 0 ≤ θ ≤ 360
42.
Solve 2 cos 2 θ + 5cos θ − 3 =
0 for 0 ≤ θ ≤ 360
43.
Solve cos 2 x + 3sin x − 2 =
0 for 0 ≤ x ≤ 2π
44.
Solve
45.
Solve 1 − sin θ =
3 cos θ for 0 ≤ θ ≤ 2π
46.
Find all radian solutions for 2sin 2 4θ − 2 cos 4θ =
1
47.
Solve triangle ABC given ∠B =
57 , ∠C =
31 , and side a = 7.3 meters.
48.
A man standing near a radio station antenna observes that the angle of elevation to the top
of the antenna is A. He then walks a distance s further away and observes that the angle of
elevation to the top of the antenna is B. Find the height of the antenna to the nearest foot.
Assume A = 63°, B = 44° and s = 120 feet.
3 cscθ + 2 cot θ =
0 for 0 ≤ θ ≤ 360
49.
How many triangles ABC satisfy the following conditions? ∠A =
140 , b = 87 ft., and
a = 62 ft.
50.
In triangle ABC, ∠A =
27 , b = 48 cm, and a = 39 cm. Find angle B.
51.
In triangle ABC, if a = 20 m, b = 30 m, and c = 40 m, find the measure of the smallest
angle.
52.
Two planes leave an airport at the same time. Their speeds are 360 mph and 420 mph and
the angle between their courses is 28 . How far apart are they after 1.5 hours?
53.
Express the polar equation r 2 = 6sin 2θ in rectangular form.
54.
Write the pair −2 3, −2 in polar coordinates.
55.
Sketch the graph of r= 2 + 4 cos θ
(
)