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. 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:

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

Calculation Steps

To calculate a trigonometric function:

1. Identify the angle θ in degrees.
2. Convert the angle to radians if necessary (θ in radians = θ in degrees × π/180°).
3. Apply the appropriate trigonometric function.
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°).