The Trigonometric Functions
 What about angles greater than 90°? 180°?
 The trigonometric functions are defined in terms of a
point on a terminal side
 r is found by using the Pythagorean Theorem:
r x y
2
2
The 6 Trigonometric Functions of
angle  are:
sin  
r
csc   , y  0
y
cos  
r
sec   , x  0
x
x
cot   , y  0
y
tan  
, x0
sin  
y
r
The Trigonometric Functions
The trigonometric values do not depend
on the selected point – the ratios will be
the same:
First Quadrant:
sin  = +
cos  = +
tan  = +
csc  = +
sec  = +
cot  = +
Second Quadrant:
sin  = +
cos  = tan  = csc  = +
sec  = cot  = -
Third Quadrant:
sin  = cos  = tan  = +
csc  = sec  = cot  = +
y
x
Fourth Quadrant:
sin  = cos  = +
tan  = csc  = sec  = +
cot  = -
y
x
All Star Trig Class
 Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in
different quadrants:
Star
All
Sine is positive All functions
are positive
Trig
Tan is positive
Class
Cos is positive
So, now we know the signs of the trig
functions, but what about their values?...
The value of any trig function of an angle
 is equal to the value of the
corresponding trigonometric function of
its reference angle, except possibly for
the sign. The sign depends on the
quadrant that  is in.
Reference Angles
 The reference angle, α, is the angle between the
terminal side and the nearest x-axis:
All Star Trig Class
 Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in
different quadrants:
Star
All
Sine is positive All functions
are positive
Trig
Tan is positive
Class
Cos is positive
Quadrantal Angles
(terminal side lies along an axis)
Trig values of quadrantal angles:

0°
90°
180°
270°
360°
sin 
0
1
0
–1
0
cos 
1
0
–1
0
1
tan 
0
undefined
0
undefined
0
cot 
undefined
0
undefined
0
undefined
sec 
1
undefined
–1
undefined
1
undefined
1
undefined
–1
undefined
csc 
Trigonometric Identities
Reciprocal Identities
1
sin x 
csc x
1
cos x 
sec x
1
tan x 
cot x
Quotient Identities
sin x
tan x 
cos x
cos x
cot x 
sin x
Trigonometric Identities
Pythagorean Identities
The fundamental Pythagorean
identity:
sin 2 x  cos 2 x  1

Divide the first by
:
1  cot x  csc x

Divide the first by cos2x :
tan 2 x  1  sec 2 x
sin2x
2
2