Problem 32 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 450^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \sin 450^{\circ} \) is 1.

Step 1: Reduce the Angle

The first step is to reduce the given angle to an equivalent angle between 0° and 360° because the sine function is periodic. Here we reduce 450° by subtracting 360° (one full circle) until the angle falls within this range. So, 450° - 360° = 90°.

Step 2: Apply the Sine Function

Now that we have reduced the angle to 90°, we apply the sine function. We know from the unit circle that \( \sin 90^{\circ} = 1 \).

Step 3: State the Exact Value

Therefore, the exact value of \( \sin 450^{\circ} \) is 1.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is usually written as \( \sin \theta \), where \( \theta \) represents an angle. This function is essential in both mathematics and real-world applications. It helps us understand oscillations, waves, and circular movements.

Some important features of the sine function include:

• It ranges between -1 and 1 for all angles.
• It is periodic, meaning it repeats its values in regular intervals.
• Its graph is a smooth wave that oscillates above and below the horizontal axis.

Understanding the sine function's behavior is crucial for working with trigonometric equations and modeling periodic phenomena.

Periodicity

Periodicity refers to the repeating nature of the sine function. The sine function has a period of \( 360^{\circ} \) or \( 2\pi \) radians. This means every time you add 360° (or a full circle's worth of rotation) to an angle, the sine value remains the same.

This property allows us to reduce angles that are out of the standard 0° to 360° range. For example, when calculating \( \sin 450^{\circ} \), we can subtract 360° to bring the angle to 90°, which is within the usual cycle. This reduction does not change the sine value due to its periodic nature.

The periodicity of the sine function simplifies many calculations and helps with understanding patterns in circular and wave-like motion.

Unit Circle

The unit circle is an essential tool in trigonometry that helps in understanding the sine, cosine, and other trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.

When you place angles on the unit circle, their terminal sides will intersect the circle. The \( y \)-coordinate of this intersection point gives the sine value of the angle.

Some key points to remember about the unit circle:

• At \( 0^{\circ} \) and \( 360^{\circ} \), the y-coordinate is 0, so \( \sin 0^{\circ} = 0 \).
• At \( 90^{\circ} \), the y-coordinate is 1, hence \( \sin 90^{\circ} = 1 \).
• At \( 180^{\circ} \), the y-coordinate returns to 0, making \( \sin 180^{\circ} = 0 \).
• Finally, at \( 270^{\circ} \), the y-coordinate is -1, thus \( \sin 270^{\circ} = -1 \).

The unit circle provides a visual method to understand trigonometric values and their relationships, helping bridge the gap between abstract concepts and spatial reasoning.