91Ó°ÊÓ

Understanding the concept of a reference angle is crucial when working with trigonometric functions. A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis. The reference angle must always be less than 90°. This allows us to use simpler trigonometric values common to the first quadrant.
For example, if an angle like 285° is given, you can find the reference angle by subtracting it from the nearest complete circle, which is 360° for angles in degrees. Thus, the reference angle for 285° is calculated as:

• 360° - 285° = 75°

This reference angle helps in determining the exact trigonometric values, simplifying complex questions into more easily manageable parts.

Quadrants in a coordinate system help define the sign of the trigonometric function values. Angles are divided into four quadrants:

• Quadrant I: 0° to 90°
• Quadrant II: 90° to 180°
• Quadrant III: 180° to 270°
• Quadrant IV: 270° to 360°

Each quadrant has particular characteristics:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, others negative.
• Quadrant III: Tangent is positive, others negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

For the angle 285°, it's in the fourth quadrant. In this quadrant, the sine function is negative, as is confirmed by the solution to \( \sin 285° = -\sin 75° \). This knowledge is essential for determining the correct sign of trigonometric values across different angles.

Exact trigonometric values refer to the specific values of sine, cosine, and tangent for certain common angles. These values are often derived from known geometric shapes and angles, such as 30°, 45°, and 60°, and their equivalents in other quadrants.
For instance, \( \sin 75° \) can be found using trigonometric identities or tables and is precisely given by:

• \( \sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4} \)

When solving problems like \( \sin 285° \), this understanding becomes critical. Since 285° is in the fourth quadrant, where the sine function is negative, you adjust the value of \( \sin 75° \) to match the sign of the quadrant. Thus:

• \( \sin 285° = -\frac{\sqrt{6} + \sqrt{2}}{4} \)

This reflects the importance of both reference angles and quadrant signs in accurately determining exact values.