https://file.aiquickdraw.com/mathbot/file/1726154903941zq6z9jpq.jpg

The equation you provided is:

\[
\cos(2x) + (\sin(x) + \cos(x))^2 = 0
\]
for \( 0^\circ \leq x \leq 360^\circ \).

### Step 1: Expand the equation

First, expand \( (\sin(x) + \cos(x))^2 \):

\[
(\sin(x) + \cos(x))^2 = \sin^2(x) + 2\sin(x)\cos(x) + \cos^2(x)
\]

Using the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \), this simplifies to:

\[
(\sin(x) + \cos(x))^2 = 1 + 2\sin(x)\cos(x)
\]

### Step 2: Rewrite the equation

Now substitute this back into the original equation:

\[
\cos(2x) + 1 + 2\sin(x)\cos(x) = 0
\]

### Step 3: Apply the double-angle identity

Recall that \( \cos(2x) = \cos^2(x) - \sin^2(x) \) and also \( 2\sin(x)\cos(x) = \sin(2x) \), so the equation becomes:

\[
\cos(2x) + 1 + \sin(2x) = 0
\]

### Step 4: Solve the trigonometric equation

Rearranging the terms:

\[
\cos(2x) + \sin(2x) = -1
\]

This is a standard form that can be solved using various techniques, including a substitution or graphical approach, depending on the context.

Would you like further steps to solve this equation, or additional details on solving it graphically? 

### 5 Relative Questions
1. What are the general solutions for \(\cos(2x)\) and \(\sin(2x)\)?
2. How do we solve trigonometric equations using identities?
3. What is the graphical interpretation of \(\cos(2x) + \sin(2x) = -1\)?
4. What is the effect of the domain \(0^\circ \leq x \leq 360^\circ\) on the solution set?
5. How can you simplify trigonometric equations using angle sum identities?

### Tip:
When solving trigonometric equations, always try to simplify using known identities like double-angle or sum-to-product identities to make the equation more manageable.

Hence, solve cos(2x) + (sin(x) + cos(x))^2 = 0 for 0° ≤ x ≤ 360°.

Learn how to solve the trigonometric equation cos(2x) + (sin(x) + cos(x))^2 = 0 for x between 0° and 360°. This problem involves the use of double-angle and Pythagorean identities. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation