Apply compound-angle identities Compound-angle identities express the sine, cosine, and tangent of sums and differences in terms of the trigonometric ratios of the component angles, revealing how angle addition changes these quantities. The learner applies the identities to exact-value evaluation, symbolic simplification, and verification of equivalent expressions, while attending to sign changes and the restricted domains of tangent; in particular, trigonometric functions are not generally distributive over addition. This scope excludes complex-number, hyperbolic, and more abstract generalized treatments.
Detailed Explanation: Apply compound-angle identities To apply a compound-angle identity, rewrite an angle as a sum or difference of two familiar angles, such as 30 ∘ 30^\circ 3 0 ∘ 45 ∘ 45^\circ 4 5 ∘ 60 ∘ 60^\circ 6 0 ∘
The main identities are
sin ( A + B ) = sin A cos B + cos A sin B \sin(A+B)=\sin A\cos B+\cos 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 \cos(A+B)=\cos A\cos B-\sin A\sin B cos ( A + B ) = cos A cos B − sin A sin B tan ( A + B ) = tan A + tan B 1 − tan A tan B \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} tan ( A + B ) = 1 − tan A tan B tan A + tan B For differences, change the signs between the terms appropriately:
sin ( A − B ) = sin A cos B − cos A sin B \sin(A-B)=\sin A\cos B-\cos 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 \cos(A-B)=\cos A\cos B+\sin A\sin B cos ( A − B ) = cos A cos B + sin A sin B tan ( A − B ) = tan A − tan B 1 + tan A tan B \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B} tan ( A − B ) = 1 + tan A tan B tan A − tan B Example: Find the exact value of sin 75 ∘ \sin 75^\circ sin 7 5 ∘
Step 1: Rewrite the angle as a sum of familiar angles.
75 ∘ = 45 ∘ + 30 ∘ 75^\circ=45^\circ+30^\circ 7 5 ∘ = 4 5 ∘ + 3 0 ∘ So,
sin 75 ∘ = sin ( 45 ∘ + 30 ∘ ) \sin75^\circ=\sin(45^\circ+30^\circ) sin 7 5 ∘ = sin ( 4 5 ∘ + 3 0 ∘ ) Step 2: Use the sine addition identity.
sin ( A + B ) = sin A cos B + cos A sin B \sin(A+B)=\sin A\cos B+\cos A\sin B sin ( A + B ) = sin A cos B + cos A sin B Therefore,
sin ( 45 ∘ + 30 ∘ ) = sin 45 ∘ cos 30 ∘ + cos 45 ∘ sin 30 ∘ \sin(45^\circ+30^\circ)
=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ sin ( 4 5 ∘ + 3 0 ∘ ) = sin 4 5 ∘ cos 3 0 ∘ + cos 4 5 ∘ sin 3 0 ∘ Step 3: Substitute the exact trigonometric values.
= ( 2 2 ) ( 3 2 ) + ( 2 2 ) ( 1 2 ) =\left(\frac{\sqrt2}{2}\right)\left(\frac{\sqrt3}{2}\right)
+\left(\frac{\sqrt2}{2}\right)\left(\frac12\right) = ( 2 2 ) ( 2 3 ) + ( 2 2 ) ( 2 1 ) Step 4: Simplify.
= 6 4 + 2 4 =\frac{\sqrt6}{4}+\frac{\sqrt2}{4} = 4 6 + 4 2 Thus,
sin 75 ∘ = 6 + 2 4 \boxed{\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}} sin 7 5 ∘ = 4 6 + 2 Remember that trigonometric functions are not generally distributive over addition. For example,
sin ( A + B ) ≠ sin A + sin B . \sin(A+B)\ne \sin A+\sin B. sin ( A + B ) = sin A + sin B . You must use the appropriate compound-angle identity.