If sin a = 0.396 and sin ß= 0.501 with both angles' terminal rays in Quadrant-I, find the values of (a) cos(a + B) = (b) cos(a - b) Your answers should be accurate to 4 decimal places. =
If sin a = 0.396 and sin ß= 0.501 with both angles' terminal rays in Quadrant-I, find the values of (a) cos(a + B) = (b) cos(a - b) Your answers should be accurate to 4 decimal places. =
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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![### Trigonometric Problem Solving
Given the scenario where:
\[ \sin \alpha = 0.396 \]
\[ \sin \beta = 0.501 \]
with both angles’ terminal rays in Quadrant-I, find the values of:
#### (a)
\[ \cos(\alpha + \beta) \]
\[ \boxed{\ \ \ \ \ \ \ \ \ \ \ } \]
#### (b)
\[ \cos(\alpha - \beta) \]
\[ \boxed{\ \ \ \ \ \ \ \ \ \ \ } \]
**Note:** Your answers should be accurate to 4 decimal places.
### Explanation
To solve for \( \cos(\alpha + \beta) \) and \( \cos(\alpha - \beta) \), we use the angle addition and subtraction formulas for cosine:
\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
First, we need to determine \( \cos \alpha \) and \( \cos \beta \) using the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
So,
\[ \cos \alpha = \sqrt{1 - \sin^2 \alpha} \]
\[ \cos \beta = \sqrt{1 - \sin^2 \beta} \]
Given that both angles are in Quadrant-I, \( \cos \alpha \) and \( \cos \beta \) will be positive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b559ab3-5119-482d-94d5-a98a530a62e8%2Fa962cea1-39c4-4937-9470-7fd43737286f%2Fw69zhiq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Problem Solving
Given the scenario where:
\[ \sin \alpha = 0.396 \]
\[ \sin \beta = 0.501 \]
with both angles’ terminal rays in Quadrant-I, find the values of:
#### (a)
\[ \cos(\alpha + \beta) \]
\[ \boxed{\ \ \ \ \ \ \ \ \ \ \ } \]
#### (b)
\[ \cos(\alpha - \beta) \]
\[ \boxed{\ \ \ \ \ \ \ \ \ \ \ } \]
**Note:** Your answers should be accurate to 4 decimal places.
### Explanation
To solve for \( \cos(\alpha + \beta) \) and \( \cos(\alpha - \beta) \), we use the angle addition and subtraction formulas for cosine:
\[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \]
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
First, we need to determine \( \cos \alpha \) and \( \cos \beta \) using the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
So,
\[ \cos \alpha = \sqrt{1 - \sin^2 \alpha} \]
\[ \cos \beta = \sqrt{1 - \sin^2 \beta} \]
Given that both angles are in Quadrant-I, \( \cos \alpha \) and \( \cos \beta \) will be positive.
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