https://file.aiquickdraw.com/mathbot/file/172727558674248ige7sa.jpg

From the problem statement, we are given:

\[ \sin \alpha + \sin \beta = a \]
\[ \cos \alpha + \cos \beta = b \]

To find \(\cos(\alpha - \beta)\), we can use the identity:

\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]

However, first we need to find expressions for \(\cos \alpha \cos \beta\) and \(\sin \alpha \sin \beta\).

### Step 1: Square both equations
\[ (\sin \alpha + \sin \beta)^2 = a^2 \]
\[ (\cos \alpha + \cos \beta)^2 = b^2 \]

### Step 2: Expand the squares using trigonometric identities
Using the identities for the sum of trigonometric functions:
\[ \sin^2 \alpha + 2 \sin \alpha \sin \beta + \sin^2 \beta = a^2 \]
\[ \cos^2 \alpha + 2 \cos \alpha \cos \beta + \cos^2 \beta = b^2 \]

### Step 3: Simplify using \(\sin^2 + \cos^2 = 1\)
Assuming \(\alpha\) and \(\beta\) are angles in the same function domain, then:
\[ \sin^2 \alpha + \sin^2 \beta = 1 - \cos^2 \alpha + 1 - \cos^2 \beta = 2 - (\cos^2 \alpha + \cos^2 \beta) \]

Therefore:
\[ 2 - (1 - \sin^2 \alpha + 1 - \sin^2 \beta) = a^2 \]
\[ 2 - (2 - (b^2 - 2 \cos \alpha \cos \beta)) = a^2 \]

Rearranging terms, we find:
\[ 1 + \cos \alpha \cos \beta = \frac{b^2 + a^2}{2} \]
\[ \cos \alpha \cos \beta = \frac{b^2 + a^2}{2} - 1 \]

### Step 4: Use sum-to-product identities
From the sum-to-product identities:
\[ 2 \sin \alpha \sin \beta = a^2 - (2 - b^2 + 2 \cos \alpha \cos \beta) \]
\[ 2 \sin \alpha \sin \beta = a^2 - 2 + b^2 - (b^2 + a^2 - 2) \]
\[ \sin \alpha \sin \beta = \frac{a^2 + b^2 - 2}{2} \]

### Step 5: Combine to find \(\cos(\alpha - \beta)\)
\[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \]
\[ \cos(\alpha - \beta) = \left(\frac{b^2 + a^2}{2} - 1\right) + \left(\frac{a^2 + b^2 - 2}{2}\right) \]
\[ \cos(\alpha - \beta) = a^2 + b^2 - 2 \]

This is the expression for \(\cos(\alpha - \beta)\) in terms of \(a\) and \(b\). Would you like more details or do you have any further questions?

Here are some related questions you might find interesting:
1. What would be the value of \(\sin(\alpha - \beta)\) given the same conditions?
2. How would you determine \(\alpha + \beta\) using \(a\) and \(b\)?
3. Can you derive expressions for \(\tan(\alpha - \beta)\) from these conditions?
4. How would the solution change if \(\alpha\) and \(\beta\) were angles in different quadrants?
5. How can you verify the correctness of the expression for \(\cos(\alpha - \beta)\)?

**Tip:** Always check if \(\alpha\) and \(\beta\) satisfy the basic trigonometric identity \(\sin^2 x + \cos^2 x = 1\) when working on problems involving multiple angle formulas. This can help identify errors in algebraic manipulation or assumptions.

If sin α + sin β = a and cos α + cos β = b, find cos(α - β) in terms of a and b.

Learn how to derive the expression for cos(α - β) in terms of a and b given sin α + sin β = a and cos α + cos β = b. This problem involves trigonometric identities and is suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation