---### Understanding Trigonometric Identities: Sum of Sine Functions**Problem Statement:**76. Explain why \( \sin 3^\circ + \sin 357^\circ = 0 \).#### Explanation:To understand why \( \sin 3^\circ + \sin 357^\circ \) equals zero, we must explore some trigonometric identities and properties:1. **Sine Function Identity for Negative Angles:** \[ \sin(-\theta) = -\sin(\theta) \]2. **Sine Function Periodicity:** Sine function has a periodicity of \(360^\circ\), meaning: \[ \sin(\theta + 360^\circ) = \sin(\theta) \]3. **Angle Difference:** Notice that \( 357^\circ \) is equivalent to: \[ 357^\circ = 360^\circ - 3^\circ \]Using these properties, we can rewrite \( \sin 357^\circ \) as:\[ \sin 357^\circ = \sin (360^\circ - 3^\circ) \]From the identity \( \sin(360^\circ - \theta) = -\sin(\theta) \), we get:\[ \sin (360^\circ - 3^\circ) = -\sin(3^\circ) \]Therefore:\[ \sin 357^\circ = -\sin(3^\circ) \]Now, substituting this back into the original expression:\[ \sin 3^\circ + \sin 357^\circ = \sin 3^\circ + (-\sin 3^\circ) \]Simplifying this, we get:\[ \sin 3^\circ - \sin 3^\circ = 0 \]Thus, the given equation \( \sin 3^\circ + \sin 357^\circ = 0 \) is validated by the identity.---This explanation showcases the concepts of angle transformation in trigonometry and demonstrates how trigonometric identities help in simplifying expressions.

Name: ---### Understanding Trigonometric Identities: Sum of Sine Functions*
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: ---### Understanding Trigonometric Identities: Sum of Sine Functions**Problem Statement:**76. Explain why \( \sin 3^\circ + \sin 357^\circ = 0 \).#### Explanation:To understand why \( \sin 3^\circ + \sin 357^\circ \) equals zero, we must explore som