Download Math 112 Exponent and Logarithm Test

Pre-Calculus 2015-2016
Name
Math 112 Final Review
Power Standard 7—Trigonometric Functions
1.
A water wheel has a radius of 18 feet. The wheel is rotating at 15 revolutions per minutes.
Find the linear speed, in feet per second, of the water.
2.
A large pizza from Papa John’s has a radius of 9”.
a) It is cut into eight equal pieces. What angle, in degrees and convert to radians, will be
formed by a slice?
b) Before the pizza was made, the crust was thrown into the air and spun a complete
rotation every ½ second. What was the speed of the outer edge of the crust in inches
per second?
3.
A tower that is 98 feet tall casts a shadow of 113 feet long. Find the angle of elevation of the
sun to the nearest degree.
4.
The angle of elevation to the top of a pole is 62°. If the distance to the pole is 47 feet, what is
the height of the pole?
5.
3
Find the exact value of cos(𝑠𝑖𝑛−1 7) without using your calculator. Show your diagram for
full credit.
Pre-Calculus 2015-2016
6.
4
Find the exact value of tan( cos−1 5).
7.
Find the exact value of each of the remaining trigonometric functions of 𝜃 if
2
sin 𝜃 = − 5 , sec 𝜃 > 0.
8.
Solve for x in the equation on the interval [0, 2𝜋). Solve algebraically and give exact values for
the answers.
a) tan 𝑥 + 1 = 0
1
b) sin 𝑥 + 2 = 0
9.
Solve cos(x) + 2 cos(x) sin(x) = 0 on the interval [0, 2π).
Pre-Calculus 2015-2016
10. Using standard notation for a triangle, solve triangle ABC if A = 30˚, a = 44, and b = 52.
11. Using standard notation solve the triangle if 𝐴 = 130˚, 𝐶 = 10˚, 𝑐 = 4.
12. Two ships leave a harbor at the same time. Ship A travels on a bearing of 191° at 10 mph to
location A. The other ship, Ship B, travels on a bearing of 65° at 12 mph to location B. How far
apart, to the nearest mile, will the ships be after three hours?
13. A surveyor needs to find the distance between two houses that are separated by a lake. She
measures angle ACB, which is found to be 15°, and then walks off the distance to each house,
80 feet and 100 feet, respectively. How far apart are the houses?
Pre-Calculus 2015-2016
Power Standard 8—Graphs of Trigonometric Functions
14. Find the amplitude, period, phase shift, and vertical shift of 𝑦 = 6 sin(8𝑥 − 𝜋) + 2. Graph the
function.
y
8
6
4
2
-π/4 -3π/16 -π/8 -π/16
x
π/16 π/8 3π/16 π/4
-2
-4
-6
-8
𝜋
15. Find the amplitude, period, phase shift, and vertical shift of 𝑦 = 2 cos(3(𝑥 − )) − 2. Graph the
3
function.
y
4
3
2
1
-2π/3 -π/2 -π/3 -π/6
-1
-2
-3
-4
x
π/6
π/3
π/2 2π/3
Pre-Calculus 2015-2016
Power Standard 9—Analytic Trigonometry
16. Simplify cos(𝑥) sec(−𝑥).
17. Simplify
csc 𝑥 tan2 𝑥
.
sec 𝑥 sin2 𝑥+sin 𝑥
1−sin(x)
18. Prove or disprove the following identity: (sec(x) − tan(x))2 = 1+sin(x).
19. Prove 𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 4 𝑥 = 𝑐𝑜𝑠 2 𝑥.
20. Prove sin 𝑥 + cos 𝑥 cot 𝑥 = csc 𝑥.
Pre-Calculus 2015-2016
21. Find all of the solutions of 2sin2 (x) − cos(x) − 1 = 0.
22. Prove cos 2𝑢 = cos2 𝑢 − sin2 𝑢
23. Find the exact value of each of the following under the given conditions:
−24 3π
π
cos(x) =
,−
< x < −π ; tan(y) = −√6, < y < π
25
2
2
a) sin(x − y) =
b) tan(𝑥 − 𝑦) =
24. Find the exact value of the expression below:
12
3
Give that cos 𝑥 = 13, for 𝑥 in Quadrant IV and tan 𝑦 = 4 for 𝑦 in Quadrant I find cos(𝑥 − 𝑦).
Pre-Calculus 2015-2016
Power Standard 10—Vector, Parametric, and Polar Coordinates
25. Let ⃑⃑⃑⃑⃑
𝐴𝐵 be determined by the points 𝐴(2, 3) and 𝐵(−1, −4). What is the component form of
⃑⃑⃑⃑⃑
𝐴𝐵?
26. Let 𝒗 be a vector from initial point 𝑃1 to terminal point 𝑃2 . Write 𝒗 in terms of the 𝒊 and 𝒋
vectors.
𝑃1 = (−8, 5) and 𝑃2 = (−3, 1)
27. Let 𝒖 = 6𝒊 − 2𝒋, and 𝒘 = −3𝒊 − 8𝒋. Find |𝒘 − 𝒖|.
28. Let 𝒖 = −2𝒊 + 3𝒋, 𝒗 = 6𝒊 − 3𝒋, and 𝒘 = −4𝒊. Find 3𝒖 − (2𝒘 − 𝒗).
29. Consider force vectors 𝑢 and 𝑣 acting on the same point. Find the resultant magnitude and
angle 𝜃𝑅 .
‖𝑢‖ = 200 pounds, 𝜃 = 29°
‖𝑣‖ = 300 pounds, 𝜃 = 205°
Pre-Calculus 2015-2016
30. The magnitude and direction of two forces acting on an object are 60 pounds, S 40˚ E, and 70
pounds N 62˚ E, respectively. Find the magnitude, to the nearest hundredth of a pound, and
the direction angle, to the nearest tenth of a degree, of the resultant force.
31. Find the horizontal and vertical components of the vector with magnitude 40 and direction
𝜃 = 20˚. Write the vector in terms of the vectors 𝒊 and 𝒋.
32. Find the measure of the angle between the vectors 𝑟 = 〈−3, 4〉 and 𝑠 = 〈6, 8〉. Round your
answer to the nearest degree. Explain why the vectors are orthogonal or not orthogonal.
33. A force of 60 pounds on a rope is used to pull a box up a ramp inclined at 20˚ from the
horizontal. The rope forms an angle of 43˚ with the horizontal. How much work is done
pulling the box 22 feet along the ramp?
Pre-Calculus 2015-2016
3𝜋
34. Given the point (𝑟, 𝜃) = (2, 4 ):
a) Plot the point in the polar coordinate system.
b) Give another representation of the point in
polar form using a negative r value.
c) Convert the point to rectangular coordinates
(x, y). (Leave as exact values)
7𝜋
35. Given the point (𝑟, 𝜃) = (−3, 12 ):
a) Plot the point in the polar coordinate system.
b) Give another representation of the point in
polar form using a negative r value.
c) Convert the point to rectangular coordinates
(x, y).