Understanding the Sine Function

What is the Sine Function?

The sine function, denoted as sin(θ), is one of the fundamental trigonometric functions. It relates the angles of a right triangle to the ratios of the lengths of its sides. In the context of the unit circle, sin(θ) represents the y-coordinate of the point where the angle's terminal side intersects the circle.

Formula and Definition

For a right-angled triangle with an angle θ, the sine function is defined as:

\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]

Where:

• "opposite" is the length of the side opposite to the angle θ
• "hypotenuse" is the length of the longest side of the right triangle

Calculation Steps

1. Identify the angle θ in degrees or radians.
2. If the angle is in degrees, convert it to radians using the formula: θ (in radians) = θ (in degrees) × π/180°.
3. Apply the sine function: sin(θ).
4. Round the result to the desired number of decimal places.

Example Calculation

Let's calculate sin(30°):

1. θ = 30°
2. Convert to radians: 30° × π/180° = π/6 radians
3. sin(π/6) = 0.5

Visual Representation

This diagram illustrates sin(30°) in the unit circle. The green line represents the sine value, which is the y-coordinate (0.5) of the point where the angle's terminal side intersects the circle.