Understanding the Tangent Function

What is the Tangent Function?

The tangent function, denoted as tan(θ), is one of the fundamental trigonometric functions. It relates the angles of a right triangle to the ratio of the lengths of its opposite and adjacent sides. In the context of the unit circle, tan(θ) represents the slope of the line from the origin to the point where the angle's terminal side intersects the circle.

Formula and Definition

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

\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}}\]

Where:

• "opposite" is the length of the side opposite to the angle θ
• "adjacent" is the length of the side adjacent to the angle θ

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 tangent function: tan(θ).
4. Round the result to the desired number of decimal places.

Example Calculation

Let's calculate tan(45°):

1. θ = 45°
2. Convert to radians: 45° × π/180° = π/4 radians
3. tan(π/4) = 1

Visual Representation

This diagram illustrates tan(45°) in the unit circle. The green line represents the tangent value, which is the y-coordinate (1) of the point where a line perpendicular to the x-axis at x=1 intersects the angle's terminal side extended.