LARTRIG10 1.7.053. Find the exact value of the expression, if possible. (If not possible, enter cos(tan-(3)) V10 10
LARTRIG10 1.7.053. Find the exact value of the expression, if possible. (If not possible, enter cos(tan-(3)) V10 10
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![### Example Problem - Trigonometry
#### Problem Statement:
**3.** *(DETAILS LARTRIG10 1.7.053)*
Find the exact value of the expression, if possible. (If not possible, enter IMP.)
\[ \cos(\tan^{-1}(3)) \]
Given the following expression:
\[ \frac{\sqrt{10}}{10} \]
#### Explanation:
To solve the expression, follow these steps:
1. Identify the inner function: \(\tan^{-1}(3)\)
2. The inverse tangent function, \(\tan^{-1}(3)\), gives an angle \(\theta\) such that \(\tan(\theta) = 3\).
3. Use a right triangle to interpret the inverse tangent. In this case, the opposite side is 3 and the adjacent side is 1 (since \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = 3\)).
4. Calculate the hypotenuse of the triangle using the Pythagorean theorem:
\[
\text{hypotenuse} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]
5. Find \(\cos(\theta)\). By definition, \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\):
\[
\cos(\theta) = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}
\]
Thus, the exact value of the expression \(\cos(\tan^{-1}(3))\) is \(\frac{\sqrt{10}}{10}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bf5cc85-f331-41fb-a902-3c0ed5d3d96d%2F0884e567-9cc9-47e4-8bc3-2358f04c4116%2Fx51xkz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem - Trigonometry
#### Problem Statement:
**3.** *(DETAILS LARTRIG10 1.7.053)*
Find the exact value of the expression, if possible. (If not possible, enter IMP.)
\[ \cos(\tan^{-1}(3)) \]
Given the following expression:
\[ \frac{\sqrt{10}}{10} \]
#### Explanation:
To solve the expression, follow these steps:
1. Identify the inner function: \(\tan^{-1}(3)\)
2. The inverse tangent function, \(\tan^{-1}(3)\), gives an angle \(\theta\) such that \(\tan(\theta) = 3\).
3. Use a right triangle to interpret the inverse tangent. In this case, the opposite side is 3 and the adjacent side is 1 (since \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = 3\)).
4. Calculate the hypotenuse of the triangle using the Pythagorean theorem:
\[
\text{hypotenuse} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]
5. Find \(\cos(\theta)\). By definition, \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\):
\[
\cos(\theta) = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}
\]
Thus, the exact value of the expression \(\cos(\tan^{-1}(3))\) is \(\frac{\sqrt{10}}{10}\).
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