Download Trig Review for Final

MAC 1114
Review for Final Exam
Definition I: Trigonometric Functions


1.
Find all six trigonometric functions of θ if the point  5 ,2 is on the terminal side of θ.
2.
Find sin  and tan  if cos  
24
and θ terminates in QIV.
25
Definition II: Right Triangle Trigonometry
3.
Find the values of the six trigonometric ratios of , where θ
is the angle adjacent to the side of length 3 and opposite the
side of length 2.
sin = _______
csc = _______
cos = _______
sec = _______
tan = _______
cot = _______
3
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 (5th Edition see page 76 figure 9; 6th Edition see page 77 figure 9) the distance from A to D
is y, the distance from D to C is x, and the distance from C to B is h. If A = 43º,
BDC  57  and y = 10, find x.
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.
10.
Find the exact values of each of the following trigonometric functions using reference angles
when necessary.
a)
sin 225
b)
cos(56)
c)
tan 300
d)
sin (72)
e)
cot (3)
f)
sec (–34)
Find θ, if 0º < θ < 360º and sin   
3
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)
sin   
1
2
b)
cos   0
c)
tan    3
12.
 1
2 
 , find
,
If angle θ is in standard position and intersects the unit circle at the point 
5
5


sinθ , cosθ and tanθ .
Arc Length and Area of a Sector
Memorize:  
s
r
A
1 2
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)
d)
18.
y  3 sin x
y  3 csc x


y  sin x  
4

b)
y  3 sin 4 x
y  1  3 sin 4 x
e)
c)
y   cos x
y   sec x


y  2 cos 3 x  
2

Sketch the graph of each equation below. Be sure to state exact values for the asymptotes
and x-intercept.
a)
1
x
2
1
y  cot x
2
y  tan
b)
y  tan 4 x
y  cot 4 x
Proving Identities
19.
Verify each identity:
a)
tan 2  cos 2   cot 2  sin 2   1
b)
sec sin 

 2 tan 
csc  cos 
c)
sin (csc   sin )  cos 2 
d)
sec2   tan 2   1
e)
1
1

 2 csc2 x
1  cos x 1  cos x
f)
 3

sin
 x    cos x
 2

g)




cos x    cos x    2 cos x
4
4


h)
cos 2 x
 cos x  sin x
cos x  sin x
i)
cot  
j)
sin 3 x  sin x
 cot 2 x
cos x  cos 3 x
sin 2
1  cos 2
Sum and Difference Formulas & Double Angles
20.
Find the exact value of each of the following under the given conditions:
sin A 
3

,....where ....0  A 
5
2
cos B 
2 5
3
,....where ....
 B  2
5
2
a) sin ( + )
b) cos ( + )
c) tan (  )
d) sin 2
e) cos 2
Half Angle Formulas
21.
Find the exact value of cos 15  .
22.
If cos B  
1
B
with B in quadrant III, then what is sin   ?
4
2
Identities and Formulas Involving Inverse Functions
In questions 23 – 24 evaluate each expression below without using a calculator. Assume any
variables represent positive numbers.
23.
3


sin sin 1  tan 1 2 
5


24.
tan(cos 1 2 x )
Solving Trigonometric Equations
25.
26.
Solve each equation on 0  x  360  .
a)
4 sin x  3  2 sin x
b)
2 tan x  2  0
c)
2 sin 2 x  sin x  1  0
d)
4 cos 2 x  1  0
e)
4 sin x  2 csc x  0
g)
sin 2 x  cos x  0
h)
4 sin 2 x  4 cos x  5  0
i)
sin 2 x 
Find all solutions if 0    2 . Use exact values only.
cos 2 cos   sin 2 sin   
3
2
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
d)
z 3i
e)
z  2  2i 3
c)
z  0  3i
f)
z  3  4i
Products and Quotients in Trigonometric Form & DeMoivre’s Theorem
In questions 39 – 40 multiply or divide as indicated. Leave your answer in trigonometric form.



39.
z1  5 cos 25   i sin 25 
40.
z1  10(cos 50   i sin 50  )
41.
Given z 1  1  i 3 and z 2   3  i

and
z 2  3 cos 40   i sin 40 
and
z 2  2(cos 20   i sin 20  )
a)
Find the product z1  z 2 in standard form.
b)
Then, write z1 and z 2 in trigonometric form.
c)
Now, find their product in trigonometric form.
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.
42.
 2 cos 150

 i sin 150 

4
43.

3i

4
Roots of a Complex Number
44.
45.
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.
Given the equation x 3  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.