Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ

검증된 단계별 안내

Start by examining the right-hand side (RHS) of the equation: \(\sin \theta - \cos^{2} \theta \sin \theta\).

Factor out \(\sin \theta\) from the RHS to simplify the expression: \(\sin \theta (1 - \cos^{2} \theta)\).

Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\), which implies \(1 - \cos^{2} \theta = \sin^{2} \theta\).

Substitute \(1 - \cos^{2} \theta\) with \(\sin^{2} \theta\) in the factored expression: \(\sin \theta \cdot \sin^{2} \theta\).

Simplify the expression to get \(\sin^{3} \theta\), which matches the left-hand side (LHS), thus verifying the identity.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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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. Verifying an identity means showing both sides simplify to the same expression, often using known identities like Pythagorean or angle sum formulas.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This fundamental relationship allows substitution between sine and cosine terms, which is essential for simplifying and verifying trigonometric expressions.

Algebraic Manipulation of Trigonometric Expressions

Simplifying trigonometric expressions often requires factoring, expanding, or rearranging terms. Recognizing patterns like factoring out common terms (e.g., sin θ) helps transform one side of the equation to match the other, confirming the identity.

Verify that each equation is an identity.sin³ θ = sin θ - cos² θ sin θ