Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Evaluate Composite Trig Functions
3:15 minutesProblem 65
Textbook Question
In Exercises 63–82, use a sketch to find the exact value of each expression. tan (cos⁻¹ 5/13)
Verified step by step guidance1
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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Video duration:
3mKey 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°).
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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.
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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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