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Trigonometric
Functions
Cosecant is reciprocal of sine.
Secant is reciprocal of cosine.
Cotangent is reciprocal of tangent.
Suppose the terminal side of a
rotation angle (K) passes through
the point (–3, 4).
Draw angle K in standard position.
Find the distance from (–3, 4) to
the origin using r  x 2  y 2 .
r 5
Evaluate the six trig functions
for this angle.
sin K  4 5
cos K  3 5
tan K  4 3
csc K  5 4
sec K  5 3
cot K  3 4
Suppose angle K measures approximately ―233.13°. Do the
calculator values for sine, cosine, and tangent of this angle
match the answers you previously found for angle K?
sin K  4 5
sin  233.13  0.8
cos K  3 5
cos  233.13  0.6
tan K  4 3
tan  233.13  1.333
A reference triangle can be formed by
drawing a perpendicular segment from a point
on the angle’s terminal side to the x-axis.
This perpendicular segment (a y-value) can
act as the “opposite” side, and its distance
from the distance from origin (an x-value)
serves as the “adjacent” side.
Here, however, the opposite and adjacent
legs can be negative.
x = “adjacent” leg
y = “opposite” leg
These values, because they are coordinates,
can be positive or negative (or even zero).
r = “hypotenuse”
The r-value, however, because it is a distance,
is defined to always be positive.
Angle: θ, Point: (12, –5)
A) Sketch the angle in standard position,
and indicate the Quadrant where the
terminal side lies.
B) Find the value of r using r 
x2  y 2 .
r  13
Quadrant IV
C) Evaluate the six trig functions for this angle.
sin   513
cos  12 13
tan   512
csc  13 5
sec  1312
cot   12 5
Angle: α, Point: (–1, 2)
A) Sketch the angle in standard position,
and indicate the Quadrant where the
terminal side lies.
B) Find the value of r using r 
x2  y 2 .
r 5
Quadrant II
C) Evaluate the six trig functions for this angle.
2 5
sin  
5
5
csc 
2
5
cos  
5
sec   5
tan   2
1
cot   
2
Angle: Z, Point: (–3, –3)
A) Sketch the angle in standard position,
and indicate the Quadrant where the
terminal side lies.
B) Find the value of r using r 
x2  y 2 .
r 3 2
Quadrant III
C) Evaluate the six trig functions for this angle.
2
sin   
2
csc   2
2
cos  
2
sec   2
tan  
1
cot  
1
Angle: P, Point: (24, 7)
A) Sketch the angle in standard position,
and indicate the Quadrant where the
terminal side lies.
B) Find the value of r using r 
x2  y 2 .
r  25
Quadrant I
C) Evaluate the six trig functions for this angle.
sin  
7
25
25
csc 
7
24
cos 
25
25
sec 
24
7
tan  
24
24
cot  
7
Based on the answers to these problems, tell in which Quadrant the
following are always true.
• Only sine (and cosecant) values are positive Quadrant II
• Only cosine (and secant) values are positive Quadrant IV
• Only tangent (and cotangent) values are positive Quadrant III
• All trig ratio values are positive Quadrant I
Label the Quadrants in which the six trig functions are positive.
sin, csc
sin, cos, tan,
cot,sec, csc
tan, cot
cos,sec
3
sin   , Quad II
5
3
A) Sketch the angle in standard position
B) Find the missing value
5
-4
x  r 2  y2 .
x  4
x’s are negative in Quad II
C) Evaluate the other trig functions for this angle.
sin  
3
5
5
csc 
3
4
cos  
5
5
sec 
4
3
tan  
4
4
cot  
3
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