Analysis Midterm Review – Topics 2013/2014
6.1 Trigonometric Functions – Angles & Their Measures (Quiz: 6.1 Angles & Their Measures)
6.2 The Sine, Cosine, and Tangent Functions (Chart and Quiz: 6.2 SOH-CAH-TOA)
Graphing Sine & Cosine (Quiz: Graphing)
7.1 Basic Trig Identities (Problem Set and Quiz: Basic Identities)
7.2-7.3 Addition & Subtraction Identities, Double Angle Identities, Half-Angle Identities (Quiz:
Identities)
Trigonometric Equations (Quiz: Trig Equations)
8.2 Right-Triangle Trig Applications (Quiz: Right Triangle Trig)
8.3 Law of Sines and 8.4 Law of Cosines (Quiz: Law of Sines/Cosines)
6.1 Trigonometric Functions – Angles & Their Measures
Find the radian measure of three angles (at least one positive and one negative) in standard
position that are coterminal with the given measure.
1.
p
2. -
6
4p
5
Find the degree measure of three angles (at least one positive and one negative) in standard
position that are coterminal with the given measure.
4. 505
3. 273
Convert the given degree measure to radians. (REDUCE)
4. 165
5. 48
Convert the given radian measure to degrees.
6.
3p
2
7. -
5p
8
6.2 The Sine, Cosine, and Tangent Functions
SOH-CAH-TOA
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
csc  
1
hypotenuse

sin 
opposite
sec  
1
hypotenuse

cos 
adjacent
cot  
1
adj

tan  opp
Coordinate Plane
Identify the exact value for the given trig function.
1. sin780 =__________
2. cos780 =__________
3. tan780 =__________
4. sin
5p
=__________
4
5. cos
5p
=__________
4
6. tan
5p
=__________
4
7. sin
11p
=__________
6
8. cos
11p
=__________
6
9. tan
11p
= __________
6
10. csc135 =_________
11. sec135 =_________
12. cot135 =_________
Assuming the terminal side of an angle in standard position lies in the given quadrant on the
given straight line. Find sine, cosine, & tangent.
2
5
13. Quadrant IV: Line with equation y = - x .
14. Quadrant III: Line with equation 2y-3x=0
15. Quadrant II: Line perpendicular to 5y-10=x.
Write the expression as a single real number. Do not use decimal approximations.
16. sin
5p
p
5p
p
cos + cos sin
4
6
4
6
17. Find: sin q and tan q if cosq = -
6
p
and £ q £ p .
10
2
18. What is the value of sine of an angle with    
3
25
(quadrant 3) whose secant is 
?
2
24
19. Find the cosecant of an angle whose terminal side in standard position passes through the
point (-5,-6).
Graphing Sine & Cosine
Graph the following sine & cosine functions. Identify the amplitude, period, midline, phase shift,
and critical points of each graph.
1. -2sin(3x)
2.  cos( x   )  2


3. 6sin  4 x 

3
2
4. Using the below graph of f (x) = -sin x , answer each of the below questions.
2
f(x) = 1∙sin(x)
1.5
1
0.5
π
4
0.5
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
1
1.5
2
2.5
a) Identify the domain of f (x) = -sin x in interval notation.
__________
b) Identify the range of f (x) = -sin x in interval notation.
__________
æ 3p ö
÷.
è 2 ø
c) Find f ç
æ 7p ö
÷
è 2 ø
d) Find f ç
______________
______________
e) Identify where f (x) = 0 for all values in the interval [ 0, 2p ] . _________
7.1 Basic Trig Identities
Basic Identities
csc  
1
sin 
sec  
cos 2   sin 2   1
1
cos 
tan  
sin 
cos 
1  tan 2   sec 2 
cot  
cos 
sin 
1  cot 2   csc 2 
Prove the following trigonometric identities. You must show ALL work to receive full credit!
1.
sec x  cot x
 csc2 x
sin x
2. cos x  cot x  sin x  csc x
3. sec 2 x cot 2 x  tan 2 x csc 2 x  sec 2 x csc 2 x
Addition & Subtraction Identities, Double Angle Identities, Half-Angle Identities
Addition and Subtraction Identities
sin( x  y )  sin x cos y  cos x sin y
sin( x  y )  sin x cos y  cos x sin y
cos( x  y )  cos x cos y  sin x sin y
cos( x  y )  cos x cos y  sin x sin y
tan( x  y ) 
tan x  tan y
1  tan x tan y
tan( x  y ) 
tan x  tan y
1  tan x tan y
Double-Angle Identities
sin 2 x  2 sin x cos x
cos 2 x  cos 2 x  sin 2 x
2 tan x
tan 2 x 
1  tan 2 x
cos 2 x  1  2 sin 2 x
cos 2 x  2 cos 2 x  1
Half-Angle Identities
sin
x
1  cos x

2
2
cos
x
1  cos x

2
2
tan
x
1  cos x

2
1  cos x
tan
x 1  cos x

2
sin x
tan
x
sin x

2 1  cos x
1. Using your addition/subtraction identities, find each of the following:
 5 
sin 
.
 12 
 5 
cos .
 12 
2. Given sin x =
-3
3p
and
< x < 2p . Find each of the following:
5
2
sin2x
cos2x
tan2x
3. Using your half-angle identities, find each of the following:
 5 
sin  
 12 
4. If csc 2 x 
 5 
cos 
 12 
 5 
tan  
 12 
5 1
, find sin 2x.
2
Trig Equations
Solve the following trigonometric equations using your knowledge of special trig values
or the unit circle. All answers should be given in radian measure.
1. sin x  
2
2
2. sec x  2
3. tan 2 x  1
2
4. 4 sin x  3  0
2
5. 4 cos x  4 cos x  1  0
csc
6.
x
1
3
8.2 Right-Triangle Trig Applications
1. A 35 foot line is used to tether a helium-filled balloon. Because of a breeze, the line makes an
angle of 68 with the ground. What is the height of the balloon?
2. A blimp flying at an altitude of 4280 feet spots a house on the ground. The angle of
depression from the blimp to the house is 24 . Assuming the ground is flat, how far must the
blimp fly to be directly over the house?
3. A ladder is resting against the side of a shed. If the ladder is 25 feet long & needs to reach 15
feet up the side of the shed, what angle will the ladder make with the ground? How far must
the base of the ladder be?
4. A water tower is located 300 feet from a building. From a window in the building, the angle of
elevation to the top of the tower is 35 . The angle of depression from the window to the
bottom of the tower is 18 .
a. How tall is the tower?
b. How high is the window?
5. Maria needs to know the height of a tree. From a given point on the ground, she finds that the
angle of elevation to the top of the tree is 25 . She then moves 35 feet closer to the tree.
From the second point, the angle of elevation to the top of the tree is 38 . Find the height of
the tree.
8.3 Law of Sines and 8.4 Law of Cosines (Quiz: Law of Sines/Cosines)
a
b
c
=
=
sin A sin B sinC
a 2 = b 2 + c 2 - 2bc cos A
b 2 = a 2 + c 2 - 2ac cos B
c 2 = a 2 + b 2 - 2ab cosC
Solve triangle ABC with the given information using the Law of Sines or the Law of Cosines.
1. a=10, b=16, A= 30 
2. B= 20.67 , C= 34 , b=185.
4. a=6, b=12, c=16.
5. a=44, c=84, C= 42.2
3. a=7, c=16, A= 30 