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 ⎟⎠