https://file.aiquickdraw.com/mathbot/file/17289569640937aqemewu.jpg

The expression in the image is:

\[
(\sin \phi + \cos \phi)^2
\]

Let’s expand and simplify it.

### Step 1: Use the identity for a binomial square  
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Here, \(a = \sin \phi\) and \(b = \cos \phi\).

### Step 2: Apply the identity to the expression  
\[
(\sin \phi + \cos \phi)^2 = \sin^2 \phi + 2\sin \phi \cos \phi + \cos^2 \phi
\]

### Step 3: Use the Pythagorean identity  
We know that:
\[
\sin^2 \phi + \cos^2 \phi = 1
\]

Thus:
\[
\sin^2 \phi + 2\sin \phi \cos \phi + \cos^2 \phi = 1 + 2\sin \phi \cos \phi
\]

### Final Simplified Expression  
\[
(\sin \phi + \cos \phi)^2 = 1 + 2\sin \phi \cos \phi
\]

This is the simplified form.

---

Do you have any questions or need further details?

### Related Questions:
1. How can the identity \(\sin^2 \phi + \cos^2 \phi = 1\) be derived?
2. What are other useful trigonometric identities?
3. Can the expression \(1 + 2\sin \phi \cos \phi\) be further factored?
4. What happens if \(\phi\) takes specific values, such as \(0^\circ\) or \(45^\circ\)?
5. How can we rewrite \(2\sin \phi \cos \phi\) in terms of a double angle?

**Tip:** The expression \(2\sin \phi \cos \phi\) can also be rewritten using the double angle identity: \(2\sin \phi \cos \phi = \sin(2\phi)\).

Learn how to simplify the expression (sin φ + cos φ)^2 using trigonometric identities. This problem involves key trigonometric concepts such as the Pythagorean identity and the double angle identity. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation