2.3
Evaluating Trigonometric
Functions for any Angle
JMerrill, 2009
Review from 2.2
Find the exact values of the other five trig functions for an
angle θ in standard position, given
o
5
sin    , 270    360o
13
12
270o
13
5
13
12
12
cos  
13
sec  
5
tan   
12
12
cot   
5
360o
θ
13
csc   
-5
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°
cos 300o
tan 225o
Negative
Positive
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
Determine the Quadrant
In which quadrant is θ if cot θ and sec θ have opposite signs?
Quadrants III and IV
Using the Sign
If
1
cos    and θ lies in Quadrant III, find sin θ and tan θ
2
3
sin   
2
-1
θ
-√3
2
tan   3
Ranges of Trigonometric
Functions
•
•
•
•
•
y
We know that sin  
r
If the measure of  increases
toward 90o, then y increases
The value of y approaches r,
and they are equal when   90o
So, y cannot be greater than r.
Using the convenient point (0,1)
y can never be greater than 1.
  90o
r
y

x
Ranges Continued
• Using a similar approach, we get:
1  sin   1
1cos   1
sec   1 or sec   1
csc   1 or csc   1
tan  and cot  can be any real number
Determining if a Value is
Within the Range
• Evaluate (calculator)
(not possible)
cos  2
cot   0
  90o
3
sin  
2
(not possible)
Reference Angles
Reference Angle: the smallest positive 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.
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
d) Find the exact value of tan 495o
To find the correct quadrant, find the smallest
positive coterminal angle. 495o - 360o = 135o
tan 495o = tan 135o. 135o is in Quad. II where
tangent is negative. The reference angle = 45o
tan 495o = - tan 45o = -1
Finding Exact Measures
of Angles
• Find all values of
 3
 , where 0    360 , when sin  
2
o
o
• Sine is negative in QIII and QIV
• 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 QIII and
QIV.
  240 and 300
o
o
Approximating
• Approximate the value of
 , if sin   .6293
• 1. Ignore the negative and do sin 1  (.6293)
 38.99849667
• 2. The answer is the reference angle, which we
will round to 39o
• 3. Sine is negative in QIII and QIV
• 4. 219o and 321o
Approximating
• Approximate the value of
• 1.
 , if sin   .6293
sin 1  (.6293)
 38.99849667
• 2. The answer is the reference angle, which we
will round to 39o
• 3. Sine is positive in QI and QII
• 4. 39o and 141o
You Do
• Find all values of
 , where 0    360 when cos   0.5299
o
o
Reference angle is 58o
122o and 238o