Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

검증된 단계별 안내

Start by recalling the identity for cosine of a difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).

Apply the identity \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \) to the formula for \( \cos(A - B) \). Substitute \( A = \frac{\pi}{2} \) and \( B = \theta \) into the formula.

This substitution gives \( \cos(\frac{\pi}{2} - \theta) = \cos \frac{\pi}{2} \cos \theta + \sin \frac{\pi}{2} \sin \theta \).

Since \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \), simplify the expression to \( \cos(\frac{\pi}{2} - \theta) = 0 \cdot \cos \theta + 1 \cdot \sin \theta = \sin \theta \).

Now, to find the addition formula for \( \sin(A + B) \), use the identity \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \) and apply it to \( \sin(A + B) = \cos(\frac{\pi}{2} - (A + B)) \). Substitute \( A + B \) into the formula for \( \cos(A - B) \) to derive \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations in trigonometry. The identity sin θ = cos (π/2 − θ) is a co-function identity that relates sine and cosine, illustrating how these functions are interconnected.

Cosine of a Difference Formula

The cosine of a difference formula states that cos(A - B) = cosA cosB + sinA sinB. This formula is essential for deriving other trigonometric identities and is used to express the cosine of the difference of two angles in terms of the sines and cosines of those angles. It serves as a foundational tool in proving various trigonometric relationships.

Sine Addition Formula

The sine addition formula states that sin(A + B) = sinA cosB + cosA sinB. This formula allows us to express the sine of the sum of two angles in terms of the sines and cosines of the individual angles. It is crucial for solving problems involving the addition of angles and is derived using the cosine of a difference formula and other trigonometric identities.