Use the given information to find the exact value of each of the following: sin 2θ
cot θ = 2, θ lies in quadrant III.

검증된 단계별 안내

Recall the double-angle identity for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).

Use the given information \(\cot \theta = 2\) to express \(\tan \theta\) as \(\frac{1}{2}\), since \(\cot \theta = \frac{1}{\tan \theta}\).

Determine the signs of \(\sin \theta\) and \(\cos \theta\) in quadrant III, where both sine and cosine are negative.

Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and the value \(\tan \theta = \frac{1}{2}\) to set up a ratio between \(\sin \theta\) and \(\cos \theta\).

Find exact values for \(\sin \theta\) and \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), then substitute these into the double-angle formula \(\sin 2\theta = 2 \sin \theta \cos \theta\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Cotangent and Its Relationship to Sine and Cosine

Cotangent (cot θ) is the reciprocal of tangent, defined as cot θ = cos θ / sin θ. Knowing cot θ allows us to express sine and cosine in terms of each other, which is essential for finding sin 2θ. Understanding this relationship helps in determining the values of sine and cosine from cotangent.

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent functions depends on the quadrant where the angle lies. Since θ is in quadrant III, both sine and cosine are negative, but tangent (and cotangent) is positive. This information is crucial to assign correct signs to sine and cosine values when calculating sin 2θ.

Double-Angle Identity for Sine

The double-angle identity for sine states that sin 2θ = 2 sin θ cos θ. This formula allows us to find sin 2θ once sin θ and cos θ are known or expressed in terms of cot θ. Applying this identity is the key step in solving the problem.

Use the given information to find the exact value of each of the following: sin 2θ cot θ = 2, θ lies in quadrant III.