If A and B are acute angles, sin A = 3/5 and cos B = 5/13 find (A) Sin(A + B) (B) Sin(A - B) (C) Cos(A+B) (D) Cos(A - B)

Given that \( A \) and \( B \) are acute angles, and \( \sin A = \frac{3}{5} \) and \( \cos B = \frac{5}{13} \), let's solve for the required trigonometric values.

### Step 1: Determine \(\cos A\) and \(\sin B\)
Since \( A \) and \( B \) are acute angles, their sine and cosine values are positive.

- **For \( A \):**
  \[
  \sin^2 A + \cos^2 A = 1
  \]
  \[
  \left(\frac{3}{5}\right)^2 + \cos^2 A = 1
  \]
  \[
  \frac{9}{25} + \cos^2 A = 1
  \]
  \[
  \cos^2 A = 1 - \frac{9}{25} = \frac{16}{25}
  \]
  \[
  \cos A = \frac{4}{5}
  \]

- **For \( B \):**
  \[
  \sin^2 B + \cos^2 B = 1
  \]
  \[
  \sin^2 B + \left(\frac{5}{13}\right)^2 = 1
  \]
  \[
  \sin^2 B + \frac{25}{169} = 1
  \]
  \[
  \sin^2 B = 1 - \frac{25}{169} = \frac{144}{169}
  \]
  \[
  \sin B = \frac{12}{13}
  \]

### Step 2: Calculate the required trigonometric expressions

(A) **\(\sin(A + B)\):**
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
Substituting the values:
\[
\sin(A + B) = \frac{3}{5} \times \frac{5}{13} + \frac{4}{5} \times \frac{12}{13} 
\]
\[
\sin(A + B) = \frac{15}{65} + \frac{48}{65} = \frac{63}{65}
\]

(B) **\(\sin(A - B)\):**
\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\]
Substituting the values:
\[
\sin(A - B) = \frac{3}{5} \times \frac{5}{13} - \frac{4}{5} \times \frac{12}{13} 
\]
\[
\sin(A - B) = \frac{15}{65} - \frac{48}{65} = \frac{-33}{65}
\]

(C) **\(\cos(A + B)\):**
\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\]
Substituting the values:
\[
\cos(A + B) = \frac{4}{5} \times \frac{5}{13} - \frac{3}{5} \times \frac{12}{13}
\]
\[
\cos(A + B) = \frac{20}{65} - \frac{36}{65} = \frac{-16}{65}
\]

(D) **\(\cos(A - B)\):**
\[
\cos(A - B) = \cos A \cos B + \sin A \sin B
\]
Substituting the values:
\[
\cos(A - B) = \frac{4}{5} \times \frac{5}{13} + \frac{3}{5} \times \frac{12}{13}
\]
\[
\cos(A - B) = \frac{20}{65} + \frac{36}{65} = \frac{56}{65}
\]

### Summary of Results
\[
\sin(A + B) = \frac{63}{65}, \quad \sin(A - B) = \frac{-33}{65}
\]
\[
\cos(A + B) = \frac{-16}{65}, \quad \cos(A - B) = \frac{56}{65}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the value of \(\tan(A + B)\) using the values of \(\sin(A + B)\) and \(\cos(A + B)\)?
2. How do the trigonometric identities change if \(A\) or \(B\) is not an acute angle?
3. How would you compute \(\sin(2A)\) and \(\cos(2B)\)?
4. Can you find the exact values of \(A\) and \(B\) using their sine and cosine?
5. How would the calculations differ if one angle was a right angle?

**Tip:** Remember that the sum and difference identities for sine and cosine can help simplify problems involving the addition or subtraction of angles.

Explore how to solve Sin(A + B) and Cos(A - B) using trigonometric identities. Given that Sin(A) = 3/5 and Cos(B) = 5/13, learn the step-by-step calculations for these trigonometric expressions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation