Compute the exact value of 11-TI tan 12 Answer:
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°.
Related questions
Question
![**Problem Statement:**
Compute the exact value of
\[
\tan\left(\frac{11 \cdot \pi}{12}\right)
\]
**Answer:**
*(This section should include the explanation or the detailed solution to find the exact value of the tangent expression.)*
**Explanation:**
To find the exact value of \(\tan\left(\frac{11 \pi}{12}\right)\), we can use the angle sum identity:
\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}
\]
Identify two angles \(A\) and \(B\) such that their sum is \(\frac{11\pi}{12}\). In this case, let's use
\(A = \frac{2\pi}{3}\) and \(B = \frac{\pi}{4}\).
Calculate \(\tan\left(\frac{2\pi}{3}\right)\) and \(\tan\left(\frac{\pi}{4}\right)\):
- \(\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}\)
- \(\tan\left(\frac{\pi}{4}\right) = 1\)
Substitute these values into the angle sum formula:
\[
\tan\left(\frac{11 \pi}{12}\right) = \frac{-\sqrt{3} + 1}{1 - (-\sqrt{3} \cdot 1)} = \frac{-\sqrt{3} + 1}{1 + \sqrt{3}}
\]
To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[
\tan\left(\frac{11 \pi}{12}\right) = \frac{(-\sqrt{3} + 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}
\]
Calculate:
- Numerator: \((- \sqrt{3} + 1)(1 - \sqrt{3}) = -\sqrt{3} + 3 - \sqrt{3} + 1 = 4 - 2\sqrt{3}\)
- Denominator: \((1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 = -2\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53dc3663-7fd1-433e-97d7-8ab208c305a6%2F19115212-23a9-45ef-a5a8-67a71d06e12f%2Fmcckrd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Compute the exact value of
\[
\tan\left(\frac{11 \cdot \pi}{12}\right)
\]
**Answer:**
*(This section should include the explanation or the detailed solution to find the exact value of the tangent expression.)*
**Explanation:**
To find the exact value of \(\tan\left(\frac{11 \pi}{12}\right)\), we can use the angle sum identity:
\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}
\]
Identify two angles \(A\) and \(B\) such that their sum is \(\frac{11\pi}{12}\). In this case, let's use
\(A = \frac{2\pi}{3}\) and \(B = \frac{\pi}{4}\).
Calculate \(\tan\left(\frac{2\pi}{3}\right)\) and \(\tan\left(\frac{\pi}{4}\right)\):
- \(\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}\)
- \(\tan\left(\frac{\pi}{4}\right) = 1\)
Substitute these values into the angle sum formula:
\[
\tan\left(\frac{11 \pi}{12}\right) = \frac{-\sqrt{3} + 1}{1 - (-\sqrt{3} \cdot 1)} = \frac{-\sqrt{3} + 1}{1 + \sqrt{3}}
\]
To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[
\tan\left(\frac{11 \pi}{12}\right) = \frac{(-\sqrt{3} + 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}
\]
Calculate:
- Numerator: \((- \sqrt{3} + 1)(1 - \sqrt{3}) = -\sqrt{3} + 3 - \sqrt{3} + 1 = 4 - 2\sqrt{3}\)
- Denominator: \((1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 = -2\
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