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Trigonometric Identities
Trigonometric Functions
adj
1
⎛π
⎞
cosθ =
=
= sin ⎜ − θ ⎟ = cos ( −θ ) ⎝2
⎠
hyp secθ
Domain
Range
θ ∈ −1 ≤ cosθ ≤ 1
sin θ =
opp
1
⎛π
⎞
=
= cos ⎜ − θ ⎟ = − sin ( −θ ) ⎝
⎠
hyp cscθ
2
θ ∈ −1 ≤ sin θ ≤ 1
tan θ =
opp
1
sin θ
⎛π
⎞
=
= cot ⎜ − θ ⎟ = − tan ( −θ ) =
⎝
⎠
adj cot θ
2
cosθ
1⎞
⎛
θ ≠ ⎜ n + ⎟ π , n ∈ ⎝
2⎠
−∞ ≤ tan θ ≤ ∞
secθ =
hyp
1
⎛π
⎞
=
= csc ⎜ − θ ⎟ = sec ( −θ ) ⎝2
⎠
adj cosθ
1⎞
⎛
θ ≠ ⎜ n + ⎟ π , n ∈ ⎝
2⎠
secθ ≤ −1 or secθ ≥ 1
cscθ =
hyp
1
⎛π
⎞
=
= sec ⎜ − θ ⎟ = − csc ( −θ ) ⎝2
⎠
opp sin θ
θ ≠ nπ , n ∈ cscθ ≤ −1 or cscθ ≥ 1
cot θ =
adj
1
cosθ
⎛π
⎞
=
= tan ⎜ − θ ⎟ = − cot ( −θ ) =
⎝2
⎠
opp tan θ
sin θ
θ ≠ nπ , n ∈ −∞ ≤ cot θ ≤ ∞
Pythagorean Identities
sin 2 θ + cos 2 θ = 1 Sum and Difference Identities
sin ( A ± B ) = sin A cos B ± cos Asin B
1+ cot 2 θ = csc 2 θ cos ( A ± B ) = cos A cos B  sin Asin B
tan 2 θ + 1 = sec 2 θ tan ( A ± B ) =
Double-Angle Identities Half-Angle Identities
A
1− cos A
sin = ±
2
2
sin 2A = 2sin A cos A cos 2A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1− 2sin 2 A tan 2A =
2 tan A
1− tan 2 A
Product-to-Sum Identities
1
cos A cos B = ⎡⎣ cos ( A − B ) + cos ( A + B ) ⎤⎦ 2
tan A ± tan B
1  tan A tan B
cos
A
1+ cos A
=±
2
2
tan
A
1− cos A 1− cos A
sin A
=±
=
=
2
1+ cos A
sin A
1+ cos A
Sum-to-Product Identities
⎛ A + B⎞
⎛ A − B⎞
cos A + cos B = 2 cos ⎜
cos ⎜
⎟
⎝ 2 ⎠
⎝ 2 ⎟⎠
sin Asin B =
1
⎡ cos ( A − B ) − cos ( A + B ) ⎤⎦ 2⎣
⎛ A + B⎞ ⎛ A − B⎞
cos A − cos B = −2sin ⎜
sin
⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠
sin A cos B =
1
⎡sin ( A + B ) + sin ( A − B ) ⎤⎦ 2⎣
⎛ A + B⎞
⎛ A − B⎞
sin A + sin B = 2sin ⎜
cos ⎜
⎝ 2 ⎟⎠
⎝ 2 ⎟⎠
cos Asin B =
1
⎡sin ( A + B ) − sin ( A − B ) ⎤⎦ 2⎣
⎛ A + B⎞ ⎛ A − B⎞
sin A − sin B = 2 cos ⎜
sin
⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠