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Chapter 10
Trigonometric Functions
Trigonometric Ratios
Special Angles
Trigonometric Ratios of any angles
The Trigonometric Functions
we will be looking at
SINE
COSINE
TANGENT
Greek Letter q
Prounounced
“theta”
Represents an unknown angle
Angle Measures and Types of
Angles
• The most common unit for measuring angles is
the degree. (One rotation = 360o)
0
o
o
1
360
rotation

1
• ¼ rotation = 90 , ½ rotation = 180 ,
• Types of angles named on basis of measure:
0o  q  90o
q  90 o
90o  q  180o
q  180o
Basic Terms continued
• Positive angle: The
rotation of the
terminal side of an
angle
counterclockwise.
• Negative angle: The
rotation of the
terminal side is
clockwise.
Opp
Sin 
Hyp
Adj
Cos 
Hyp
Opp
Tan 
Adj
hypotenuse
q
adjacent
opposite
opposite
Find the values of the three trigonometric functions of q.
?
5
4
q
Pythagorean Theorem:
(3)² + (4)² = c²
5=c
3
opp 4
adj 3
opp 4


sin q 


cos q 
tan
q
hyp 5
hyp 5
adj 3
Trigonometric Ratios of Special Angles
• The trigonometric ratios of angles measuring
30o, 45o and 60o can be obtained using a
square and an equilateral triangle.
Complementary and Supplementary
Angles
• Two positive angles are called complementary
if the sum of their measures is 90o
o
o
47
• The angle that is complementary to 43 =
• Two positive angles are called
supplementary if the sum of their measures
is 180o
• The angle that is supplementary to 68o = 112o
Positive Trig Function
Values
STUDENTS
Sine and its
reciprocal
are positive
ALL
y
-y
r
r
-x
y
All functions
are positive
x
r
TAKE
Tangent and
its reciprocal
are positive
r
-y
CALCULUS
Cosine and its
reciprocal are
positive
Positive, Negative or
Zero?
sin 240°
Negative
cos 300o
Positive
tan 225o
Positive
Determine the Quadrant
In which quadrant is θ if cos θ
and tan θ have the same sign?
Quadrants I and II
Determine the Quadrant
In which quadrant is θ if cos θ
is negative and sin θ is positive?
Quadrant II
Using the Sign
If
1
cos q   and θ lies in Quadrant III, find sin θ and tan θ
2
3
sin q  
2
-1
θ
-√3
2
tan q  3
Reference Angles
Reference Angle or Basic Angle: the smallest acute angle
determined by the x-axis and the terminal side of θ
ref angle
ref angle
ref angle
ref angle
Think of the reference angle as a “distance”—how
close you are to the closest x-axis.
Reference Angles
A reference angle is the acute angle formed by the
terminal side of
and the horizontal axis.
q
Find the reference angles of 300o and –135o.
300o
60o
45o
-135o
Find Reference Angle
150°
30°
225°
45°
300°
60°
Using Reference Angles
a) sin 330° =
= - sin 30°
= - 1/2
b) cos 120° =
= - cos 60°
=-½
Using Reference Angles
c) sin (-120°)=
= - sin 60°

3
2
Finding Exact Measures
of Angles
• Find all values of
 3
q , where 0  q  360 , when sin q 
2
o
o
• Sine is negative in Q III and Q IV
• Using the 30-60-90 values we found
earlier, we know
3
o
sin 60 
2
Finding Exact Measures
of Angles – Cont.
•
3
sin 60 
2
o
• Our reference angle is 60o. We must be
60o off of the closest x-axis in Q III and
QIV.
q  240 and 300
o
o