In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ cot θ = 3, θ lies in quadrant III.

검증된 단계별 안내

insert step 1: Understand that \( \cot \theta = 3 \) implies \( \tan \theta = \frac{1}{3} \) because \( \tan \theta = \frac{1}{\cot \theta} \).

insert step 2: Since \( \theta \) is in quadrant III, both sine and cosine are negative, but tangent is positive.

insert step 3: Use the double angle identity for tangent: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).

insert step 4: Substitute \( \tan \theta = \frac{1}{3} \) into the double angle identity: \( \tan 2\theta = \frac{2 \times \frac{1}{3}}{1 - (\frac{1}{3})^2} \).

insert step 5: Simplify the expression to find \( \tan 2\theta \).

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주요 개념

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the double angle formula for tangent, which states that tan(2θ) = 2tan(θ) / (1 - tan²(θ)). Understanding these identities is crucial for simplifying and solving trigonometric expressions.

Quadrants and Signs of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant III, both sine and cosine are negative, which means tangent (the ratio of sine to cosine) is positive. Knowing the quadrant in which the angle lies helps determine the signs of the trigonometric values needed for calculations.

Finding Trigonometric Values from Cotangent

Cotangent is the reciprocal of tangent, defined as cot(θ) = 1/tan(θ). Given cot(θ) = 3, we can find tan(θ) as 1/3. This relationship allows us to derive other trigonometric values, such as sine and cosine, using the Pythagorean identity, which is essential for calculating tan(2θ) accurately.