Rewrite the following as a single expression using only sine or only cosine. sin(x) cos(x) + cos(r) sin(1 x) = ¯
Rewrite the following as a single expression using only sine or only cosine. sin(x) cos(x) + cos(r) sin(1 x) = ¯
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
![**Rewrite the following as a single expression using only sine or only cosine.**
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) = \_\_\_\_\_\_ \]
To solve this problem, we can use the sum-to-product identities in trigonometry. Specifically, we use the sine addition formula:
\[ \sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B) \]
So, for our given expression:
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) \]
We can identify \( A = \frac{9}{11}x \) and \( B = \frac{13}{41}x \), and apply the formula:
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) = \sin\left(\frac{9}{11}x + \frac{13}{41}x\right) \]
Thus, the single expression using only sine is:
\[ \sin\left(\frac{9}{11}x + \frac{13}{41}x\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2F4cb80a9f-a128-4457-88d2-4535e6e1e55e%2Fxea93tf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Rewrite the following as a single expression using only sine or only cosine.**
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) = \_\_\_\_\_\_ \]
To solve this problem, we can use the sum-to-product identities in trigonometry. Specifically, we use the sine addition formula:
\[ \sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B) \]
So, for our given expression:
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) \]
We can identify \( A = \frac{9}{11}x \) and \( B = \frac{13}{41}x \), and apply the formula:
\[ \sin\left(\frac{9}{11}x\right) \cos\left(\frac{13}{41}x\right) + \cos\left(\frac{9}{11}x\right) \sin\left(\frac{13}{41}x\right) = \sin\left(\frac{9}{11}x + \frac{13}{41}x\right) \]
Thus, the single expression using only sine is:
\[ \sin\left(\frac{9}{11}x + \frac{13}{41}x\right) \]
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