Problem 96 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \sin 0 $$

Step-by-Step Solution

Answer

The exact value of \( \sin 0 \) is 0.

Step 1: Understanding Sine Function

The sine function, denoted as \( \sin \theta \), is a trigonometric function that calculates the ratio of the length of the opposite side to the hypotenuse in a right-angle triangle. For angles measured in radians or degrees, it can also be interpreted on the unit circle.

Step 2: Locate the Angle on the Unit Circle

The angle \( 0 \) radians corresponds to the point \((1, 0)\) on the unit circle. This is where the angle starts from the positive x-axis.

Step 3: Determine the Sine Value

On the unit circle, the sine of an angle is represented by the y-coordinate of the corresponding point. For \( \sin 0 \), the coordinate is \((1, 0)\) where the y-coordinate is 0.

Step 4: Conclusion and Result

Since the y-coordinate of the point \((1, 0)\) is 0, the exact value of \( \sin 0 \) is 0.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions that plays a crucial role in mathematics, especially in topics involving triangles and circulatory motion. It is denoted by \( \sin \theta \), where \( \theta \) represents an angle. This function essentially tells you how far "up" the angle takes you from the x-axis on a standard unit circle.
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. This idea can be expanded when dealing with angles not necessarily confined to triangles by using the unit circle.
Some important points about the sine function include:

• It is periodic with a cycle repeating every \( 2\pi \) radians or 360 degrees.
• Its values range from \(-1\) to \(1\).
• It is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).

Unit Circle Approach

A powerful and visual method to understand trigonometric functions like sine is by using the unit circle approach. The unit circle is a circle with a radius of 1 centered on the origin of a coordinate plane. This circle helps establish a clear visual representation of how angles correspond to different points.
When you draw an angle starting from the positive x-axis, the point where the corresponding terminal side intersects the unit circle gives you coordinates \((x, y)\). These coordinates are essential in determining the trigonometric functions:

• The x-coordinate gives the cosine value.
• The y-coordinate gives the sine value.

For the angle of 0 radians, you start from the positive x-axis, so the coordinates are \((1, 0)\). This means that at 0 radians or 0 degrees, the sine value is directly observable as the y-coordinate, which in this case is 0.

Exact Values of Trigonometric Functions

In trigonometry, it becomes important to recognize and determine exact values of trigonometric functions like sine, cosine, and tangent for specific angles. Often, these values are needed for angles that are commonly encountered such as 0, 30, 45, 60, and 90 degrees or their equivalent in radians.
For these key angles, instead of using a calculator to get approximate decimals, you can rely on known exact values:

• \( \sin 0 = 0 \)
• \( \sin \frac{\pi}{6} = \sin 30^{\circ} = \frac{1}{2} \)
• \( \sin \frac{\pi}{4} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin \frac{\pi}{3} = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{2} = \sin 90^{\circ} = 1 \)

Mastering these values allows you to quickly solve problems without the need for computational tools and helps deepen understanding of how these functions behave geometrically.