In a triangle ABC if sin^2 A + sin^2 B = sin^2 C, then the value of angle C is?

Given the equation \( \sin^2 A + \sin^2 B = \sin^2 C \) in a triangle \( ABC \), we need to find the value of angle \( C \).

To solve this, let's use the sine rule and some trigonometric identities.

### Step-by-Step Solution:

1. **Recall the Sine Rule:**
   The sine rule in a triangle states that:
   \[
   \sin A + \sin B + \sin C = 180^\circ
   \]
   Hence, \( C = 180^\circ - (A + B) \).

2. **Substitute and Simplify:**
   Given \( \sin^2 A + \sin^2 B = \sin^2 C \), substitute \( C = 180^\circ - (A + B) \). We know from trigonometry:
   \[
   \sin(180^\circ - \theta) = \sin \theta
   \]
   Therefore, \( \sin C = \sin(180^\circ - (A + B)) = \sin(A + B) \).

   Substituting into the equation:
   \[
   \sin^2 A + \sin^2 B = \sin^2 (A + B)
   \]

3. **Using the Identity for \( \sin^2 (A + B) \):**
   \[
   \sin^2 (A + B) = \sin^2 A \cos^2 B + \cos^2 A \sin^2 B + 2 \sin A \sin B \cos A \cos B
   \]
   This simplifies to:
   \[
   \sin^2 A + \sin^2 B = (\sin^2 A + \sin^2 B)(1 - \sin^2 A \sin^2 B) + 2 \sin A \sin B \cos A \cos B
   \]

4. **Using the Pythagorean Identity:**
   The identity states \( \sin^2 A + \sin^2 B + \cos^2 C = 1 \). If \( A + B + C = 180^\circ \), then:
   \[
   C = 90^\circ
   \]

   Substituting this into the equation:
   \[
   \sin^2 A + \sin^2 B = \sin^2(90^\circ) = 1
   \]

5. **Conclusion:**
   If \( \sin^2 A + \sin^2 B = \sin^2 C \) and using the properties and identities of triangles, the condition is satisfied when \( C = 90^\circ \).

Hence, **the value of angle \( C \) is \( 90^\circ \)**.

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Would you like to see more details on any step or have any questions?

### Related Questions:

1. What is the value of \( A \) and \( B \) if \( C = 90^\circ \) and \( \sin^2 A + \sin^2 B = 1 \)?
2. How can we use trigonometric identities to solve equations involving angles in triangles?
3. How does the sine rule apply in non-right triangles?
4. Can the above problem be solved using cosine instead of sine?
5. What are the applications of the Pythagorean trigonometric identity in solving problems?

### Tip:

For triangles, always remember that the sum of the angles is \(180^\circ\), and the trigonometric identities can simplify complex expressions involving angles.

Learn how to find angle C in triangle ABC using trigonometric identities and the sine rule. Understand the relationship between sine squares of angles and their cosine counterparts. Suitable for high school students studying trigonometry.

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