Final Exam Review 1 - Scf

MAC 1114 Review for Final Exam
Definition I: Trigonometric Functions
1. Find all six trigonometric functions of θ if the point ( 5,
2)
2. Find sin θ and tan θ if
Definition II: Right Triangle Trigonometry
24
cos θ = and θ terminates in QIV.
25
3. Find the values of the six trigonometric ratios of θ, where θ
is the angle adjacent to the side of length 3 and opposite the
side of length 2.
sin θ = _______ csc θ = _______
cos θ = _______ sec θ = _______
− is on the terminal side of θ.
tan θ = _______ cot θ = _______ 2
Solving Right Triangles
4. In triangle ABC with C = 90˚, A = 10˚42´ and b = 5.932 cm solve for the missing parts of the
triangle.
5. In the figure (textbook 6 th Edition see page 77 figure 9) the distance from A to D is y, the distance from
o
D to C is x, and the distance from C to B is h. If A = 43º, ∠BDC = 57 and y = 10, find x.
3

Applications Involving Right Triangles
6. A ship is offshore New York City. A sighting is taken of the Statue of Liberty,which is about 305
feet tall. If the angle of elevation to the top of the statue is 20°, how far is the ship from the base
of the statue? (Round to nearest hundredth)
7. A boat travels on a course of bearing N 37º 10′ W for 79.5 miles. How many miles north and
how many miles west has the boat traveled?
8. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters
from the base. What angle does the cable form with the vertical tower?
Reference Angles
9. Find the exact values of each of the following trigonometric functions using reference angles when
necessary.
a) sin 225° b) cos(5π/6) c) tan 300°
d) sin (7π/2) e) cot (−3π) f) sec (–3π/4)
3
10. Find θ, if 0º < θ < 360º and sinθ = − and θ is in QIV.
2
Definition III: Circular Functions
11. Use the unit circle to find all values of θ between 0 and 2π for which
a)
1
sinθ = −
b) cos θ = 0
c) tanθ = − 3
2

12.
⎛
If angle θ is in standard position and intersects the unit circle at the point ⎜
⎝
cosθ and tanθ .
1
, −
5
2 ⎞
⎟ , find sinθ ,
5 ⎠
Arc Length and Area of a Sector
Memorize:
s
θ =
θ
r
2 1
A = r
2
13. Find the length of an arc of a circle of radius 3 feet if the arc subtends a central angle of 30°.
Also, find the area of this sector.
14. A person standing on the earth notices that a 747 Jumbo Jet flying overhead subtends an angle of
0.45º. If the length of the jet is 230 feet, find its altitude to the nearest thousand feet.
15. Find the area of the sector formed by central angle θ in a circle of radius r if
θ = 15º and r = 5 m.
16. A lawn sprinkler is located at the corner of a yard. The sprinkler is set to rotate through 90º and
project out 60 feet. What is the area of the yard watered by the sprinkler?

Graphs of the Trigonometric Functions
17. Find the amplitude, period and phase shift. Then, sketch one complete period of each graph. On
your graphs state the exact values for the endpoints, quarter, half and three-quarter points of the
period.
a) y = 3sin
x
b) y = 3sin
4x
c) y = −cosπx
y = 3csc
x
y = −1+
3sin
4x
y = −secπx
⎛ π ⎞
⎛ π ⎞
d) y = sin⎜
x − ⎟ e) y = 2cos⎜3x
+ ⎟
⎝ 4 ⎠
⎝ 2 ⎠
18. Sketch the graph of each equation below. Be sure to state exact values for the asymptotes and xintercept.
a)
1
y = tan x
2
b) y = tan 4x
1
y = cot x
2
y =
cot 4x

Proving Identities
19. Verify each identity:
2 2
2 2
secθ
sinθ
a) tan θ cos θ + cot θ sin θ = 1
b) + = 2 tanθ
cscθ
cosθ
2
2
2
c) sin θ (cscθ
− sinθ
) = cos θ
d) sec θ − tan θ = 1
1 1
2
e) + = 2csc
x
1+
cos x 1−
cos x
⎛ π ⎞ ⎛ π ⎞
g) cos ⎜ x + ⎟ + cos⎜
x − ⎟ = 2 cos x
⎝ 4 ⎠ ⎝ 4 ⎠
i)
⎛ 3π
⎞
f) sin⎜ − x⎟ = −cos
x
⎝ 2 ⎠
cos 2x
h) = cos x − sin x
cos x + sin x
sin 2θ
sin 3x
−
sin x
cotθ
= j) = cot 2x
1−
cos 2θ
cos x − cos3x

Sum and Difference Formulas & Double Angles
20. Find the exact value of each of the following under the given conditions:
3
π
sin A = ,.... where....
0 < A <
5
2
2 5
3π
cos B = ,.... where....
< B < 2π
5
2
a) sin (Α + Β)
b) cos (Α + Β)
c) tan (Α + Β)
d) sin 2Α
e) cos 2Α
Half Angle Formulas
21. Find the exact value of
22. If
o
cos 15 .
1
⎛ ⎞
cos B = − with B in quadrant III, then what issin⎜ ⎟
4
⎝ 2 ⎠
B ?
Identities and Formulas Involving Inverse Functions
In questions 23 – 24 evaluate each expression below without using a calculator. Assume any variables
represent positive numbers.
⎛ −1
3 −1
⎞
23. sin⎜sin
− tan 2⎟
⎝ 5 ⎠
24. tan(cos 2 )
1 −
x

Solving Trigonometric Equations
25. Solve each equation on
0 <
o
≤ x 360 .
a) 4 sin x − 3 = 2sin
x
b) 2 tan x + 2 = 0
2
2
c) 2sin
x − sin x −1
= 0
d) 4cos
x −1
= 0
e) 4 sin x − 2csc
x = 0
g) sin 2x
− cos x = 0
2
h) 4sin
x + 4cos
x − 5 = 0
i)
26. Find all solutions if 0 ≤ θ < 2π
. Use exact values only.
cos 2θ
cosθ
− sin 2θ
sinθ
= −
3
2
sin 2x
=
3
2

Law of Sines
27. In triangle ABC if A = 24.7º , C = 106.1º , and b = 34.0 cm, find the missing parts.
28. In triangle ABC use the law of sines to show that no triangle exists for which A = 60º , a = 12
inches and b = 42 inches.
29. In triangle ABC use the law of sines to show that exactly one triangle exists for which
A = 42º, a = 29 inches, and b = 21 inches.
30. Find two triangles ABC and A ′ B′
C′
for which A = 51º , a = 6.5 feet, and b = 7.9 feet. State the
measure of all three angles in each of the two triangles.
31. A man standing near a building notices that the angle of elevation to the top of the building is 64º.
He then walks 240 feet farther away from the building and finds the angle of elevation to the top
to be 43º. Find h, the height of the building?
h
240 feet
32. A hot-air balloon is held at a constant altitude by two ropes that are anchored to the ground. One
rope is 120 feet long and makes an angle of 65 with the ground. The other rope is 115 feet long.
What is the distance between the points on the ground at which the two ropes are anchored?

Law of Cosines
33. In triangle ABC if C = 120º, a = 10 cm, and b = 12 cm, use the law of cosines to find c.
34. In triangle ABC if a = 5 km, b = 7 km, and c = 9 km, use the law of cosines to find C to the nearest
tenth of a degree.
35. The diagonals of a parallelogram are 26.8 meters and 39.4 meters. If they meet at an angle of
134.5º, find the length of the shorter side of the parallelogram.
36. If a = 50 yd, b = 75 yd, and c = 65 yd, then what is the area of triangle ABC?
37. Use Heron’s Formula to find the area of the triangle sides 7 meters, 15 meters and 12 meters.
Trigonometric Form for Complex Numbers
38. Write each number in trigonometric form using degrees for your angles.
a) z = 2 − 2i
b) z = −3
+ 0i
c) z = 0 − 3i
d) z = − 3 + i e) z = −2
− 2i
3 f) z =
3 − 4i

Products and Quotients in Trigonometric Form & DeMoivre’s Theorem
In questions 39 – 40 find the product 1 2 z z ⋅ and the quotient
z
form.
o
o
o
o
39. z = ( cos25
+ isin
25 ) and z = 3(
cos40
+ isin
40 )
1
5 2
o
o
o
o
40. z = (cos50
+ isin
50 ) and z = 2(cos20
+ isin
20 )
1
10 2
41. Given z = 1+
i 3 and z = − 3 + i
1
a) Find the product 1 2 z z ⋅ in standard form.
2
b) Then, write z 1 and z 2 in trigonometric form.
c) Now, find their product in trigonometric form.
z 1
. Leave your answer in trigonometric
d) Convert the answer that is in trigonometric form to standard form to show that the two
products are equal.
In questions 42– 43 use Demoivre’s Theorem to find each of the following . Write your answer in
standard a + bi form.
[ ] 4
o
o
42. 2(
cos150
+ isin150
)
43. ( ) 4
−
3 + i
2

Roots of a Complex Number
44. Given z = −4
3 + 4i
a) Write z in trigonometric form.
b) Find the three cube roots of z. Leave your answers in trigonometric form.
3
45. Given the equation x + 8 = 0
a) Solve for x by factoring and then using the quadratic formula.
b) Solve for x by finding the three cube roots of –8 in trigonometric form.
c) Convert the answer that is in trigonometric form to standard form to show that the
solutions are equal.