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Some Trigonometric Identities.
Negative Angle Identities
sin(-  ) = - sinθ
cos(-  ) = cosθ
tan(-  ) = - tanθ
csc(-  ) = - cscθ
sec(-  ) = secθ
cot(-  ) = - cotθ
Addition and Subtraction Identities
sin (A + B) =
sin A cos B + cos A sin B
cos (A + B) =
cos A cos B – sin A sin B
sin (A – B) =
sin A cos B – cos A sin B
cos (A – B) =
cos A cos B + sin A sin B
Co-function Identities If A + B = π2
sin A = cos B
sec A = csc B
tan A = cot B
Quotient Identities
1
sin 
1
secθ =
cos 
cscθ =
tanθ
=
cotθ =
1
tan 
=
Pythagorean Identities
2
2
sin θ + cos θ = 1
Double Angle Identities
sin 2 θ
= 2sinθ cosθ
cos 2θ
= cos θ -- sin 
2
sin
2
2
Half-Angle Identities
= 2cos θ -- 1
cos
tan

2

2

2
=
±
1 – cos 
2
=
±
1 + cos 
2
=
±
1 – cos 
1 + cos 
= 1 – 2sin2θ
7/02
UT Learning Center, JES A332A, 512-471-3614
cos 
sin 
THE UNIT CIRCLE
(0,1)

( ,
1
2
( ,
2
2
( , )
1
2
3
2
2
2
)
3
2
3
4
2
)

2
3
1
2
( ,
3

3
2
)
2
2
4
5
6

( , )
3
2
6
3
2
1
2
1
2
0 (1,0)
(- 1,0) 
( ,
2
2
( , )
)
7
6
( , )
2
2
2
2
11
6
5
4
7
4
4
3
( ,
1
2
3
2
)
5
3
( ,
1
2
3
2
3
2
( , )
3
2
1
2
( , )
2
2
2
2
)
(0,- 1)
sin  
TRIGONOMETRIC DEFINITIONS
SPECIAL ANGLE TRIANGLES
y
opposite

r hypotenuse
2
x
adjacent
cos   
r hypotenuse
tan  
y opposite

x adjacent

7/02
1
30°
1
3
or if it is the unit circle,
r=1 so
45°
60°
1
2
2
1
2
30°
45°
y
60°
2
1
45°
1
r
45°
2
2
x
UT Learning Center, JES A332A, 512-471-3614
3
2