Trigonometric Functions

What are Trigonometric Functions?

Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are essential in various fields, including mathematics, physics, engineering, and navigation. The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Formulas and Definitions

For a right-angled triangle with an angle θ, the trigonometric functions are defined as follows:

• \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin \theta}{\cos \theta}\)
• \(\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}\)
• \(\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}\)
• \(\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}\)

Calculation Steps

To calculate a trigonometric function:

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 appropriate trigonometric function using a calculator or mathematical software.
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 a 30-60-90 triangle in the unit circle, visually representing sin(30°) and cos(30°). The sine of 30° is the length of the opposite side (0.5), while the cosine of 30° is the length of the adjacent side (√3/2 ≈ 0.866).