Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 65

Chapter 2, Problem 65

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

Verified step by step guidance

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 \( \tan(\cos^{-1}(\frac{5}{13})) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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