Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos( π/2 + x) = -sin x

검증된 단계별 안내

Recall the angle addition formula for cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).

Apply the formula to the left side of the equation with \(a = \frac{\pi}{2}\) and \(b = x\): \(\cos\left(\frac{\pi}{2} + x\right) = \cos\frac{\pi}{2} \cos x - \sin\frac{\pi}{2} \sin x\).

Substitute the known values of \(\cos\frac{\pi}{2} = 0\) and \(\sin\frac{\pi}{2} = 1\) into the expression: \(0 \cdot \cos x - 1 \cdot \sin x\).

Simplify the expression to get \(-\sin x\) on the left side.

Since the right side of the original equation is \(-\sin x\), both sides are equal, verifying the identity.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They allow the transformation and simplification of expressions, such as rewriting cos(π/2 + x) in terms of sine or cosine functions to verify equivalences.

Angle Addition Formulas

Angle addition formulas express trigonometric functions of sums of angles, like cos(a + b) = cos a cos b - sin a sin b. These formulas are essential for breaking down complex expressions, such as cos(π/2 + x), into simpler components to facilitate verification of identities.

Relationship Between Sine and Cosine

Sine and cosine functions are closely related, often differing by phase shifts. For example, cos(π/2 + x) equals -sin x, showing how a cosine function shifted by π/2 relates directly to sine. Understanding this relationship helps in recognizing and proving trigonometric identities.

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)cos( π/2 + x) = -sin x