Question

Find the exact value of each expression. Do not use a calculator. $$ \sin 510^{\circ} $$

Short answer

\( \sin 510° = \frac{1}{2} \)

Step-by-step solution

- Understand the Angle

Notice that the angle given, 510°, is larger than 360°. To find the equivalent angle within one full rotation (0° to 360°), we need to subtract 360° from 510°.

- Subtract 360°

Subtract 360° from 510° to find the equivalent angle: \(510° - 360° = 150°\). So, \(510°\) is equivalent to \(150°\).

- Use Known Values for Sine

We know from trigonometric values that \(\sin 150°\) is related to \(\sin 30°\). Since \(150°\) is in the second quadrant where sine is positive, \(\sin 150° = \sin (180° - 30°) = \sin 30°\).

- Recall \(\sin 30°\)

From trigonometric tables, \(\sin 30° = \frac{1}{2}\). Therefore, \(\sin 150° = \frac{1}{2}\).

- Conclude the Expression

Since \(\sin 510°\) is equivalent to \(\sin 150°\), the exact value of \(\sin 510°\) is \(\frac{1}{2}\).

Key concepts

Angle Conversion

Converting angles larger than 360° into a more manageable form is crucial in trigonometry. An angle of 510° might seem challenging at first, but we can find an equivalent angle between 0° and 360°. This process helps simplify the problem. To convert 510° to a standard position angle, subtract 360°, the measure of a full rotation.

For example:
\(510° - 360° = 150°\).
Hence, 510° and 150° are co-terminal angles. This conversion makes it easier to apply trigonometric functions to find the exact value.

Sine Function

The sine function is one of the primary trigonometric functions, fundamental in both right-triangle and unit circle trigonometry. When dealing with angles, the sine function relates the angle to the ratio of the opposite side to the hypotenuse in a right triangle.
For any given angle \(\theta\), the sine function is represented as:
\(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
Using the unit circle, sine represents the y-coordinate of the corresponding point on the circle. Knowing common values, like \(\text{sin} 30° = \frac{1}{2}\), is essential for solving trigonometric problems without using a calculator.

Reference Angles

Reference angles help simplify finding trigonometric values by reducing angles to their acute counterparts. These angles are always between 0° and 90°.
Given an angle, the reference angle is the smallest angle the terminal side makes with the x-axis. For instance, the reference angle for 150° can be found as follows:
\(180° - 150° = 30°\).
This means \(\text{sin} 150° = \text{sin} 30°\). Knowing your reference angles is handy, especially when dealing with functions in different quadrants.

Trigonometric Values

Recognizing and using known trigonometric values is critical. These often involve angles like 0°, 30°, 45°, 60°, and 90°.
For example:

• \(\text{sin} 30° = \frac{1}{2}\)
• \(\text{cos} 45° = \frac{\text{sqrt}(2)}{2}\)
• \(\text{tan} 60° = \text{sqrt}(3)\)

When you're given an angle like 510° and convert it, knowing that \(\text{sin} 150° = \text{sin} 30° = \frac{1}{2}\) directly helps. Memorizing these fundamental values streamlines calculations and broadens your understanding of trigonometric functions.