Find the exact value of the expression. cos 10° cos 50° - sin 10° sin 50°
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
Find the exact value of the expression:
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
### Solution
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ = \boxed{\quad} \]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
To solve \( \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \), we can utilize the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Given \( A = 10^\circ \) and \( B = 50^\circ \), we substitute them into the formula:
\[ \cos(10^\circ + 50^\circ) = \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
\[ \cos 60^\circ = \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
Since \(\cos 60^\circ = \frac{1}{2}\):
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ = \frac{1}{2} \]
Therefore, the exact value of the expression is:
\[ \boxed{\frac{1}{2}} \]
This problem demonstrates the use of trigonometric identities to simplify expressions involving angles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F812a333f-5569-4734-822a-5aed0505f779%2F68d5d7f5-538a-42bc-b62b-393d64bddda0%2F8m0mm7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the exact value of the expression:
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
### Solution
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ = \boxed{\quad} \]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
---
### Explanation:
To solve \( \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \), we can utilize the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Given \( A = 10^\circ \) and \( B = 50^\circ \), we substitute them into the formula:
\[ \cos(10^\circ + 50^\circ) = \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
\[ \cos 60^\circ = \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ \]
Since \(\cos 60^\circ = \frac{1}{2}\):
\[ \cos 10^\circ \cos 50^\circ - \sin 10^\circ \sin 50^\circ = \frac{1}{2} \]
Therefore, the exact value of the expression is:
\[ \boxed{\frac{1}{2}} \]
This problem demonstrates the use of trigonometric identities to simplify expressions involving angles.
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