Download Pre-Calculus 12A Section 4.3 Trigonometric Ratios Coordinates in

Pre-Calculus 12A
Section 4.3
Trigonometric Ratios
Coordinates in Terms of Primary Trigonometric Ratios
P ( ) =(x,y) is a point on the terminal arm of angle  that intersects the unit circle.
PRIMARY TRIGONOMETRIC RATIOS
sin  
opp y
 y
hyp 1
y
cos  
adj x
 x
hyp 1
tan  
opp y

adj x
x
P( θ )=(cos θ,sin θ )
P ( )  (cos  ,sin ) holds true for any point P ( ) which is at the intersection of the terminal arm of angle  and
the unit circle.
You can determine the trigonometric ratios for any angle
in standard position using the coordinates of the point
where the terminal arm intersects the circle.
1
Sine 
csc  
1
Cosine
sec  
Cosecant  =
RECIPROCAL TRIGONOMETRIC RATIOS
Secant  =
Example 1: Determine the Trigonometric
Ratios for Angles in the Unit Circle
Cotangent  =
1
Tangent 
cot  
1
y
1
x
x
y
 12 5 
The point A 
,  lies at the intersection of the unit circle and the terminal arm of an angle  in standard
 13 13 
position.
a. Draw a diagram to model the situation.
b. Determine the values of the six trigonometric ratios for  . Express your answers in lowest terms.
Solution:
a.
b.
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Pre-Calculus 12A
Section 4.3
Example 2: Exact Values for Trigonometric Ratios
Exact values for the trigonometric ratios can be determined by using the unit circle or special triangles.
Determine the exact value for each of the following. Draw diagrams to illustrate your answers.
a. sin
Solution:
5
6
b. cos
2
3
c. csc315
d. tan180
Example 3: Approximate Values of Trigonometric Ratios
You can determine approximate values for sine, cosine and tangent using a calculator. Remember to set your
calculator to either degree or radian setting, depending on the question. When working with negative angles
apply your knowledge of the CAST rule and reference angles to check that your answer is reasonable.
To find the value of a trigonometric ratio for cosecant, secant or cotangent use the appropriate reciprocal
relationship.
1
sin 4.1
 1.2220
csc 4.1 
(Set calculator to radians)
Determine the approximate value for each trigonometric ratio. Give your answers to four decimal places.
a.
tan
9
5
b. sin 220
c. cos1.25
d. sec( 110)
Solution:
a.
tan
9
5
b. sin 220
c.
cos1.25
d. sec( 110)
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Pre-Calculus 12A
Section 4.3
Example 4: Determining Exact Values for Trigonometric Expressions
Determine the exact value for each of the following trigonometric expressions.
a.
sin45 cos 45  sin30sin60
 3 
2  5 
2sin2 
  cos 

 4 
 6 
b.
 2 
cos 

 3 
c.
3cos180  sin135
sin30
Solution:
a.
sin45 cos 45  sin30sin60
 3 
2  5 
2sin2 
  cos 

4 
6 


b.
 2 
cos 

 3 
c.
3cos180  sin135
sin30
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Pre-Calculus 12A
Section 4.3
Example 5: Find Angles Given Their Trigonometric Ratios
Determine the measures of all angles that satisfy the following. Use diagrams in your explanation.
a. cos   0.598472 in the domain 0    2 . Give your answers to the nearest tenth of a radian.
b. sin   0.819152 in the domain 0    360 . Give your answers to the nearest tenth of a degree.
 2
c. cos  
in the domain 0    4 . Give exact answers.
2
1
d. tan  
in the domain 180    180 . Give exact answers.
3
2
e. csc   
in the domain 2     . Give exact answers.
3
Solution:
a.
cos   0.598472 , 0    2
b. sin   0.819152 , 0    360
c.
cos  
 2
, 0    4
2
4
Pre-Calculus 12A
d. tan  
e.
csc   
Section 4.3
1
3
, 180    180
2
3
, 2    
Example 6: Calculating Trigonometric Values For Points Not On the Unit Circle
The point A(6,-8) lies on the terminal arm of an angle  in standard position. What is the exact value of each
trigonometric ratio for  .
Solution:
A(6,-8)
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