Find the exact value of the trigonometric expression without the use of a calculator. 5л 5л (3) 12 4 12 os COS 5л 12 COS COS + sin + sin 5л 12 sin 4 ... sin 4 4 Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.
Find the exact value of the trigonometric expression without the use of a calculator. 5л 5л (3) 12 4 12 os COS 5л 12 COS COS + sin + sin 5л 12 sin 4 ... sin 4 4 Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.
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
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**Title: Calculation of Trigonometric Expression**
**Objective:**
Find the exact value of the trigonometric expression without the use of a calculator.
**Expression:**
\[ \cos\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{4}\right) \]
**Step-by-Step Solution:**
1. **Identify the trigonometric identity:**
The given expression is in the form of the cosine addition formula:
\[
\cos(A) \cos(B) + \sin(A) \sin(B) = \cos(A - B)
\]
2. **Apply the identity to the given expression:**
\[
\cos\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{5\pi}{12} - \frac{\pi}{4}\right)
\]
3. **Simplify the argument of the cosine function:**
\[
\frac{5\pi}{12} - \frac{\pi}{4} = \frac{5\pi}{12} - \frac{3\pi}{12} = \frac{2\pi}{12} = \frac{\pi}{6}
\]
4. **Find the exact value:**
\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\]
**Conclusion:**
The exact value of the trigonometric expression is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]
---
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F812a333f-5569-4734-822a-5aed0505f779%2Fcdd4a73c-7c14-424f-9e13-e4d5d3b97523%2Fkq63avj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Title: Calculation of Trigonometric Expression**
**Objective:**
Find the exact value of the trigonometric expression without the use of a calculator.
**Expression:**
\[ \cos\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{4}\right) \]
**Step-by-Step Solution:**
1. **Identify the trigonometric identity:**
The given expression is in the form of the cosine addition formula:
\[
\cos(A) \cos(B) + \sin(A) \sin(B) = \cos(A - B)
\]
2. **Apply the identity to the given expression:**
\[
\cos\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{5\pi}{12} - \frac{\pi}{4}\right)
\]
3. **Simplify the argument of the cosine function:**
\[
\frac{5\pi}{12} - \frac{\pi}{4} = \frac{5\pi}{12} - \frac{3\pi}{12} = \frac{2\pi}{12} = \frac{\pi}{6}
\]
4. **Find the exact value:**
\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\]
**Conclusion:**
The exact value of the trigonometric expression is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]
---
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
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