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

검증된 단계별 안내

Recall the double-angle identity for cosine: \(\cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta}\) or alternatively \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). We will use the identity involving cotangent to find \(\cos 2\theta\).

Given \(\cot \theta = 2\), express \(\tan \theta\) as the reciprocal: \(\tan \theta = \frac{1}{2}\).

Since \(\theta\) lies in quadrant III, both sine and cosine are negative, but tangent (and cotangent) is positive, which matches \(\tan \theta = \frac{1}{2}\). Use this to find \(\sin \theta\) and \(\cos \theta\) by considering a right triangle or using the Pythagorean identity.

Set \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{1}{2}\). Let \(\sin \theta = k\) and \(\cos \theta = 2k\). Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to solve for \(k\).

Once \(\sin \theta\) and \(\cos \theta\) are found (with correct signs for quadrant III), substitute them into the double-angle formula \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\) to find the exact value of \(\cos 2\theta\).

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주요 개념

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Cotangent and its Relationship to Sine and Cosine

Cotangent (cot θ) is the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, cot θ = cos θ / sin θ. Knowing cot θ allows us to find sine and cosine values by expressing them in terms of cotangent and using the Pythagorean identity.

Double-Angle Formula for Cosine

The double-angle formula for cosine states that cos 2θ = cos² θ − sin² θ, which can also be written as 2 cos² θ − 1 or 1 − 2 sin² θ. This formula helps find the exact value of cos 2θ once sine or cosine of θ is known.

Sign of Trigonometric Functions in Quadrants

The quadrant in which angle θ lies determines the signs of sine and cosine. In quadrant III, both sine and cosine are negative. This information is crucial for correctly determining the values of sine and cosine from cotangent and for applying the double-angle formula accurately.

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