### Mathematics Exercise: Trigonometric Functions #### Question: Evaluate the following using exact degree measure. Show all of your work. \[ \sec^{-1} \left( \frac{2\sqrt{3}}{3} \right) \] #### Multiple Choice Answers: - [ ] 60° - [ ] 150° - [ ] 30° - [ ] 120° #### Explanation: To solve the given problem, recall that the secant function is the reciprocal of the cosine function. Therefore, \(\sec \theta = \frac{1}{\cos \theta}\). Given: \[ \sec \theta = \frac{2\sqrt{3}}{3} \] Then: \[ \cos \theta = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \] We need to find the angle \(\theta\) whose cosine is \(\frac{\sqrt{3}}{2}\). Checking standard angles: - \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) - \(\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\) So, the correct answer from the given options is: \[ \boxed{30^\circ} \]
### Mathematics Exercise: Trigonometric Functions #### Question: Evaluate the following using exact degree measure. Show all of your work. \[ \sec^{-1} \left( \frac{2\sqrt{3}}{3} \right) \] #### Multiple Choice Answers: - [ ] 60° - [ ] 150° - [ ] 30° - [ ] 120° #### Explanation: To solve the given problem, recall that the secant function is the reciprocal of the cosine function. Therefore, \(\sec \theta = \frac{1}{\cos \theta}\). Given: \[ \sec \theta = \frac{2\sqrt{3}}{3} \] Then: \[ \cos \theta = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \] We need to find the angle \(\theta\) whose cosine is \(\frac{\sqrt{3}}{2}\). Checking standard angles: - \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) - \(\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\) So, the correct answer from the given options is: \[ \boxed{30^\circ} \]
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Mathematics Exercise: Trigonometric Functions
#### Question:
Evaluate the following using exact degree measure. Show all of your work.
\[ \sec^{-1} \left( \frac{2\sqrt{3}}{3} \right) \]
#### Multiple Choice Answers:
- [ ] 60°
- [ ] 150°
- [ ] 30°
- [ ] 120°
#### Explanation:
To solve the given problem, recall that the secant function is the reciprocal of the cosine function. Therefore, \(\sec \theta = \frac{1}{\cos \theta}\).
Given:
\[ \sec \theta = \frac{2\sqrt{3}}{3} \]
Then:
\[ \cos \theta = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]
We need to find the angle \(\theta\) whose cosine is \(\frac{\sqrt{3}}{2}\). Checking standard angles:
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\)
So, the correct answer from the given options is:
\[ \boxed{30^\circ} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10310b6e-de2b-4f57-beaf-163189cb6f16%2Fb38ef2c4-9395-44c5-b509-9c9e112201a8%2Fdd447y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematics Exercise: Trigonometric Functions
#### Question:
Evaluate the following using exact degree measure. Show all of your work.
\[ \sec^{-1} \left( \frac{2\sqrt{3}}{3} \right) \]
#### Multiple Choice Answers:
- [ ] 60°
- [ ] 150°
- [ ] 30°
- [ ] 120°
#### Explanation:
To solve the given problem, recall that the secant function is the reciprocal of the cosine function. Therefore, \(\sec \theta = \frac{1}{\cos \theta}\).
Given:
\[ \sec \theta = \frac{2\sqrt{3}}{3} \]
Then:
\[ \cos \theta = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]
We need to find the angle \(\theta\) whose cosine is \(\frac{\sqrt{3}}{2}\). Checking standard angles:
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}\)
So, the correct answer from the given options is:
\[ \boxed{30^\circ} \]
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