Pre-Calculus 2nd Semester Review
Chapter 6
1. A photographer points a camera at a window in a nearby building forming an angle of 44 degrees
with the camera platform. If the camera is 50 meters from the building, how high above the platform is
the window?
2. Convert
17𝜋
12
into degrees.
5
3. Given tanx = − 8 and sinx > 0, find the exact value of cosx.
4. The point (8, -3) is on the terminal side of angle in standard position. Determine the exact value of
the sine of the angle.
5. Simplify:
𝑐𝑜𝑠 2 𝑥
1−𝑠𝑖𝑛2 𝑥
.
6. Find the exact value for y = sin-1(-.5)
7. Simplify sin2x + tan2x + cos2x.
8. The needle of the scale in the bulk food section of a supermarket is 20 cm long. Find the distance the
tip of the needle travels when it rotates 168°.
9. A bicycle wheel with a radius of 13 inches makes 1.1 revolutions per second. What is the speed of
the bicycle in inches per second?
10. A 95 foot long irrigation sprinkler line rotates around one end as shown. The sprinkler moves
through an arc of 130° in 1.8 hours. Find the speed of the moving end of the sprinkler in feet per
minute.
𝜋
6
11. Evaluate sec(− ). Give an exact value.
Chapter 7
1. What is the y-intercept of the graph of g(t) = cos t? Also, give the domain and range of the function.
2. Find the period, amplitude, phase shift, and vertical shift of the function y = 3 cos
3. Identify the amplitude, period, phase shift, and vertical shift of f(t) = -2 sin (
𝑡
3
4𝑥
3
.
+ 4).
4. The position of a weight attached to a spring is s(t) = -6cos(16πt) inches after t seconds. What is the
maximum height that the weight reaches above equilibrium position and when does it first reach the
maximum height?
5. Give the number of full cycles of the function that are found in the interval. Y = -sinx from [-π, 11π].
6. Give another angle that has the same value as cos
19𝜋
11
.
7. State the rule of a sine function with amplitude 8, period 5, and phase shift 14.
8. What is the domain and range of y = sec(2x)?
Chapter 8
1. Give the exact value (no calculator) sin-1(cos 0).
2. Give the exact value (no calculator) of cos-1(-1/2).
3. Find all the solutions (4) of sin (2ѳ) =
1
2
from [0, 2π). (Algebraically)
4. Find all the solutions of 5 tan2x – 13 tan x + 8 = 0 from [0, 2π). (Algebraically)
5. Solve secx = -2 for π ≤ x ≤ 3π/2. (Algebraically)
6. Find the exact solution without a calculator to tan-1(tanx) = -π/10.
7. Solve 7cscx - 4√2 = 11cscx algebraically.
8. Find all solutions to sec2x = 4tanx – 2 (exact answer where possible).
Chapter 9
1. Find the exact value of cos
𝜋
.
12
𝜋
2. Re-write in terms of sin x and cos x: sin ( 2 + 𝑥).
3. Use the half-angle identity to evaluate exactly sin
5𝜋
8
.
4. Find the exact solution of cos2 x – sin2x = ½ on the interval [0, 2π).
5. Find all solutions on the interval [0, 2π) for 2𝑠𝑖𝑛2 ∅ − 2𝑐𝑜𝑠 2 ∅ = 0.
4
6. Given tanθ=5, and 0 ≤ θ ≤ π/2 find…
a. sin2θ
θ
b. cos2
7. Express sin θ(cos2 θ) as a sum containing only sines or cosines.
8. Find sin75° exactly.
9. Prove that
1−𝑐𝑜𝑠2𝑥
𝑡𝑎𝑛𝑥
= 𝑠𝑖𝑛2𝑥.
Chapter 10
1. Solve the triangle a = 7, b = 8.6, c = 6.4.
2. Find the area of the triangle with B = 62, a = 8 and c = 3.
3. Given a triangle with A = 61, b = 18, and a = 17, find B. If there are two solutions, give both.
4. Find the area of the triangle with A = 72°, a = 13.6, and b = 13.6.
5. Solve the triangle if b = 5, c = 7, and A = 58°.
6. Two ships leave port at the same time. When ship A is 170 miles due east of the port, ship B is 200
miles from the port and 145 miles from ship A. What is ship B’s bearing?
Ship B
Port
Ship A
Chapter 11
1. Find the center, vertices, and foci of:
(𝑥+5)2
225
+
(𝑦−3)2
81
= 1.
2. Find the center, vertices, foci, and asymptotes of:
𝑥2
16
−
𝑦2
9
=1
3. Find the standard form of the parabola with vertex (0, 0) and focus at (0, -4)
4. Find the standard from of the equation 64x2 – 128x + 16y2 + 192y – 384 = 0.
5. Write an equation for the parabola with focus at (4, -9) and directrix at x = 2.
6. Write an equation for the hyperbola with foci at (8, 0) and (-8, 0) with minor axis length 8.
7. Convert (-5, 12) to the same point in the polar system.
8. Convert (−2√3, −2) into polar coordinates (exactly).
9. Write the set of parametric equations in one variable. X = 2t2, y = 3t + 1.
10. Write a parametric equation for the scenario. A banana is launched at an angle of 85° with an initial
velocity of 120 m/s at an initial height of 5 meters. Find the maximum height of the banana.
Chapter 12 Systems and Matrices
1. Solve the system using a matrix method:
x + 0y + 2z + 6w = 2
3x + 4y – 2z - w = 0
5x + 0y + 2x – 5w = -4
4x – 4y + 2z + 3w = 1
2. Given A is a 4x2 matrix and B is a 2x3 matrix, are AB and BA defined? Know how to
multiply matrices in your calculator!
x2 y 2
3. Solve algebraically y = x and

1
4 9
2
3 −7
4. Find the inverse of (
).
4 −9
5. Matt is making candles and picture frames to sell. The profit from each candle is $6 and the
profit from each picture frame is $4. He has 8 hours to make all the candles and frames he
plans to sell. It takes him 30 minutes to make each candle and 20 minutes to make each picture
frame. He has $105 for supplies. The supplies for a candle coast $3 and the supplies for a
picture frame cost $5. How many candles and picture frames should he make to maximize his
profit?
Chapter 13
1. The probability of winning a certain game is .3. If the game is played twice, what is the probability of
losing both times? What is the probability of winning once?
2. Give the expected value of the random variable with the given probability distribution
Outcome
Probability
0
0.25
1
0.25
2
0.25
3
0.25
3. How many license plates are possible with two letters followed by three digits?
4. Students are required to answer any eight out of ten questions on a certain test. How many different
combinations of eight questions can a student choose to answer?
5. The scores on a standardized college entrance exam are normally distributed. The mean score is 525,
and the standard deviation is 75. A) Of the estimated 65,000 student who took the exam, how many
scored above 600? B) What percent of the students scored below 675?
6. How many different ways can 6 different runners finish in first, second, and third places in a race.
7. Suppose two fair dice are rolled. What is the probability that a sum of 10 or 11 turns up?