Download PPT 2.3 Evaluating Trigonometric Functions for any Angle

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