Problem 13 - Algebra 2

Question

Find the exact value of each expression. \(\sin 285^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \sin 285° \) is \( -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Identify the Reference Angle

The angle given is 285°, which is more than 270° but less than 360°. To find the reference angle, we calculate: \[ 360° - 285° = 75° \] Hence, the reference angle is 75°.

Step 2: Determine the Quadrant

Since 285° is in the fourth quadrant (between 270° and 360°), we know that in this quadrant, sine is negative.

Step 3: Use the Sine of the Reference Angle

The reference angle is 75°, and using trigonometric tables or knowledge, we know:\[ \sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4} \] Since 285° is in the fourth quadrant, we have:\[ \sin 285° = -\sin 75° \]

Step 4: Calculate the Exact Value

Using the information from the previous steps, calculate the exact value:\[ \sin 285° = -\frac{\sqrt{6} + \sqrt{2}}{4} \].

Key Concepts

Reference Angle

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.

Quadrant Angles

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

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.