In Exercises 63–82, use a sketch to find the exact value of each expression. tan (cos⁻¹ 5/13)

검증된 단계별 안내

Recognize that the expression is \( \tan(\cos^{-1}(\frac{5}{13})) \). Here, \( \cos^{-1}(\frac{5}{13}) \) represents an angle \( \theta \) whose cosine is \( \frac{5}{13} \). So, set \( \theta = \cos^{-1}(\frac{5}{13}) \), which means \( \cos \theta = \frac{5}{13} \).

Draw a right triangle to represent the angle \( \theta \). Since \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13} \), label the adjacent side as 5 and the hypotenuse as 13.

Use the Pythagorean theorem to find the length of the opposite side. The formula is \( \text{opposite} = \sqrt{\text{hypotenuse}^2 - \text{adjacent}^2} = \sqrt{13^2 - 5^2} \).

Calculate the opposite side length inside the square root (do not simplify fully), so you have \( \sqrt{169 - 25} \).

Now, find \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{169 - 25}}{5} \). This expression gives the exact value of \( \tan(\cos^{-1}(\frac{5}{13})) \).

비슷한 문제에 대한 검증된 영상 답변:

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주요 개념

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

Inverse Cosine Function (cos⁻¹)

The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x. It is used to find an angle when the ratio of the adjacent side to the hypotenuse is known. The output angle lies between 0 and π radians (0° to 180°).

Right Triangle Trigonometry

Right triangle trigonometry relates the sides and angles of a right triangle. Given one angle and the hypotenuse, the other sides can be found using Pythagoras' theorem. This helps in determining other trigonometric ratios like tangent from cosine.

Tangent Function (tan)

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Once the sides are known or found, tan(θ) can be calculated as opposite/adjacent, allowing the exact value of tan(cos⁻¹(5/13)) to be determined.