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courtesy of Crafton Hills College (CHC)
Typology: Cheat Sheet
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Tangent and Cotangent Identities:
2
2
2
2
2
2
sin
= − sin 𝜃𝜃 cos
= cos 𝜃𝜃 tan
= − tan 𝜃𝜃
csc(−𝜃𝜃) = − csc 𝜃𝜃 sec(−𝜃𝜃) = sec 𝜃𝜃 cot(−𝜃𝜃) = − cot 𝜃𝜃
sin �
− 𝜃𝜃� = cos 𝜃𝜃 csc �
− 𝜃𝜃� = sec 𝜃𝜃 tan �
− 𝜃𝜃� = cot 𝜃𝜃
cos �
− 𝜃𝜃� = sin 𝜃𝜃 sec �
− 𝜃𝜃� = csc 𝜃𝜃 cot �
− 𝜃𝜃� = tan 𝜃𝜃
sin 𝛼𝛼 sin 𝛽𝛽 =
cos
− cos(𝛼𝛼 + 𝛽𝛽)
cos 𝛼𝛼 cos 𝛽𝛽 =
cos
sin 𝛼𝛼 cos 𝛽𝛽 =
[sin(𝛼𝛼 + 𝛽𝛽) + sin(𝛼𝛼 − 𝛽𝛽)]
cos 𝛼𝛼 sin 𝛽𝛽 =
[sin(𝛼𝛼 + 𝛽𝛽) − sin(𝛼𝛼 − 𝛽𝛽)]
sin 𝛼𝛼 + sin 𝛽𝛽 = 2 sin �
� cos �
sin 𝛼𝛼 − sin 𝛽𝛽 = 2 cos �
� sin �
cos 𝛼𝛼 + cos 𝛽𝛽 = 2 cos �
� cos �
cos 𝛼𝛼 − cos 𝛽𝛽 = − 2 sin �
� sin �
sin(𝛼𝛼 ± 𝛽𝛽) = sin 𝛼𝛼 cos 𝛽𝛽 ± sin 𝛽𝛽 cos 𝛼𝛼
cos(𝛼𝛼 ± 𝛽𝛽) = cos 𝛼𝛼 cos 𝛽𝛽 ∓ sin 𝛼𝛼 sin 𝛽𝛽
tan(𝛼𝛼 ± 𝛽𝛽) =
tan 𝛼𝛼 ± tan 𝛽𝛽
1 ∓ tan 𝛼𝛼 tan 𝛽𝛽
sin
= sin 𝜃𝜃 csc
= csc 𝜃𝜃
cos(𝜃𝜃 + 2 𝜋𝜋𝜋𝜋) = cos 𝜃𝜃 sec(𝜃𝜃 + 2 𝜋𝜋𝜋𝜋) = sec 𝜃𝜃
tan
= tan 𝜃𝜃 cot
= cot 𝜃𝜃
sin �
1 − cos 𝜃𝜃
cos �
1 + cos 𝜃𝜃
tan �
1 − cos 𝜃𝜃
1 + cos 𝜃𝜃
sin
= 2 sin 𝜃𝜃 cos 𝜃𝜃
cos( 2 𝜃𝜃) = cos
2
𝜃𝜃 −sin
2
= 2 cos
2
= 1 − 2 sin
2
tan 2 𝜃𝜃 =
2tan 𝜃𝜃
1 − tan
2
x
y
(𝑥𝑥, 𝑦𝑦)
r θ
hypotenuse
opposite
adjacent
sin 𝛼𝛼
=
sin 𝛽𝛽
=
sin 𝛾𝛾
2
2
2
2
2
2
2
2
tan
tan
tan
tan
tan
tan
− 1
same as
− 1
same as
− 1
𝑠𝑠𝑎𝑎𝑠𝑠𝑠𝑠 𝑎𝑎𝑠𝑠
− 1
− 1
− 1
Function Domain Range
𝑦𝑦 = sin
− 1
𝑦𝑦 = cos
− 1
𝑦𝑦 = tan
− 1
𝑦𝑦 = cot
− 1
𝑦𝑦 = sec
− 1
𝑦𝑦 = csc
− 1
sin (sin
− 1
(𝑥𝑥)) = 𝑥𝑥 sin
− 1
(sin(𝜃𝜃)) = 𝜃𝜃
cos (cos
− 1
(𝑥𝑥)) = 𝑥𝑥 cos
− 1
(cos(𝜃𝜃)) = 𝜃𝜃
tan (tan
− 1
(𝑥𝑥)) = 𝑥𝑥 tan
− 1
(tan(𝜃𝜃)) = 𝜃𝜃