MATH 114 EXAM 1 SUMMARY AND REVIEW EXAMPLES
SPRING 2010
1.
Know the unit circle in its entirety in exact radian measure and exact (x,y) coordinates.
2.
Know all basic trig identities and definitions: Pythagorean, quotient, reciprocal, cofunctions of
complementary angles, even/odd properties:
Reciprocal Identities:
csc  
1
sin 
sec  
1
cos
Quotient Identities:
tan  
sin 
cos
cot  
cos
sin 
Pythagorean Identities:
sin 2   cos2   1
Cofunction Identities:
Even-Odd Properties:
1  tan 2   sec 2 
cot  
1
tan 
1  cot 2   csc 2 


sin      cos 
2



csc     sec 
2



cos     sin 
2



sec     csc 
2



tan      cot 
2



cot      tan 
2

sin      sin 
csc     csc 
cos    cos
sec    sec
tan      tan 
cot      cot 
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
csc θ = hyp/opp
sec θ = hyp/adj
cot θ = adj/opp
For a right triangle:
3.
Know all vocabulary/concepts: standard position, unit circle, complementary, supplementary,
hypotenuse, reference angle, reference triangle, coterminal angles, cofunctions, arc length, central
angle, angle of elevation, angle of depression
4.
Know angle conversion: Degrees to radians, radians to degrees, degrees in decimal notation to
DMS notation, degrees in DMS notation to decimal notation, rotations/revolutions to degrees or
radians, radians or degrees to revolutions/rotations.
5.
Know how to apply the concepts in applications: arc length, right triangle including angle of
elevation and depression.
6.
Examples: (these are not exhaustive just sample problems)
Convert
16
to degrees. Convert 110° to radians. Convert 76.3458° to DMS notation.
45

(in radian measure). Find the supplement of 11°.
9
3
Find a positive and negative coterminal angle (in exact radian measure) of
.
Find the complement of
5
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Determine the EXACT values of the following trigonometric functions:
I.
sec
4
3
II.
sin
11
6
III.
cot
3
2
IV.
cos
7
6
V.
 3 
csc 

 4 
VII.
tan
2
3
The acute angle  has csc  
5
. Find he EXACT VALUE of the remaining five trig functions.
2
In what quadrant is the terminal side of an angle measuring -4 radians? 420°? 11 radians?
Name the quadrant in which the terminal side of angle  lies, given that cot   0 and sin   0 .
Find the reference angle of:
a) 256°
b)  200 °
c)
17
12
The point (  6,2 ) is on the terminal side of an angle α in standard position. Find the exact values of all
six trig functions.
By the Complemntary Angle Theorem, the csc 13° equals ___________ .
Without a calculator, use identities/unit circle to find EXACT value: [Note 30° =
sin 10 
cos 80 
+
 sin 30 


 sin 60




6
and 60° =

3
.]
2
=
________________
A tall tree is struck by lightning, causing a break. The part broken is still attached to the main trunk, but the
top has fallen to the ground. Forest Ranger Gisela measured the distance “x” between the trunk and the top
of the tree to be 130 feet and the angle θ made by the top with the ground to be 27° . What is the original
height of the tree? Provide ±0.01 decimal place accuracy.
The Medical Alert helicopter is flying at 720 feet. The helicopter emergency crew sees an ambulance at an
angle of depression of 58. Find the distance of the ambulance from a point on the ground directly below
the helicopter. Include a fully labeled diagram, showing angle of depression, as part of your solution.
Provide ±0.01 decimal place accuracy.
Sasha is riding on a horse on the outside rim of a merry-go-round (carousel) of diameter 60 feet. If the
angle made from the center of the merry-go-round is 142°, how far did Sasha travel?
Find the EXACT VALUES of the following trigonometric functions of  given sec  3 , and sin   0 .
[Show all work including reference triangle and/or basic trigonometric identities/definitions in general
(before putting in numerical values)]
a)
cos


b) cot   
2

d)
csc 2 
e)
cos (  )
c) sin   
f) cot θ
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