Trigonometry Calculator

How to Calculate Trigonometric Functions

Trigonometric functions are fundamental mathematical tools that relate the angles of a right triangle to the ratios of the lengths of its sides. They are widely used in various fields of mathematics, physics, and engineering.

Formulas

The six main trigonometric functions are:

• Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• Tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)} \)
• Cotangent: \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan(\theta)} \)
• Secant: \( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{1}{\cos(\theta)} \)
• Cosecant: \( \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{1}{\sin(\theta)} \)

Where:

• θ is the angle in radians
• opposite is the length of the side opposite to the angle θ
• adjacent is the length of the side adjacent to the angle θ
• hypotenuse is the length of the longest side of the right triangle

Calculation Steps

1. Ensure the angle is in radians. If it's in degrees, convert it to radians by multiplying by π/180°.
2. Identify which trigonometric function you need to calculate.
3. Use a calculator or trigonometric tables to find the value of the function for the given angle.
4. The result can be any real number, depending on the function and the angle.

Example

Let's calculate sin(30°):

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

Therefore, sin(30°) = 0.5

Visual Representation

This diagram illustrates sin(30°). The blue line represents the angle of 30°, and the red line shows the sine value of 0.5 on the y-axis.