Trigonometry Calculator

Calculate trigonometric functions for any angle

Trigonometric Values

sin(θ)

0.707107

cos(θ)

0.707107

tan(θ)

1.000000

csc(θ)

1.414214

sec(θ)

1.414214

cot(θ)

1.000000

Absolute Function Values

Common Angles Table

Angle (°)sincostan
0°010
30°0.50.8660250.57735
45°0.7071070.7071071
60°0.8660250.51.732051
90°10∞
120°0.866025-0.5-1.732051
135°0.707107-0.707107-1
150°0.5-0.866025-0.57735
180°0-10
210°-0.5-0.8660250.57735
225°-0.707107-0.7071071
270°-10∞
315°-0.7071070.707107-1
360°010

Understanding Trigonometry

What Is Trigonometry?

Trigonometry studies the relationships between angles and sides of triangles. The six trigonometric functions — sin, cos, tan, cosec, sec, cot — define ratios between sides of a right triangle based on an angle.

The Six Trigonometric Functions

For angle θ in a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. The reciprocal functions are cosec = 1/sin, sec = 1/cos, cot = 1/tan.

Unit Circle

The unit circle has radius 1 centered at the origin. Any angle θ corresponds to a point (cos θ, sin θ) on the circle. This extends trig functions to all angles, not just acute angles in triangles.

Common Angle Values

sin 0° = 0, sin 30° = 0.5, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1. These exact values appear frequently and are worth memorizing.

Applications

Navigation, surveying, physics (wave motion, oscillations), engineering (force analysis, structural calculations), computer graphics (rotations, projections), and signal processing (Fourier transforms).

Practical Example

For θ = 45°: sin(45°) = √2/2 ≈ 0.7071, cos(45°) = √2/2 ≈ 0.7071, tan(45°) = 1. The complementary angle relationship: sin(θ) = cos(90° − θ).

Perguntas Frequentes

What is the unit circle?

A circle with radius 1 centered at the origin. Any point on it is (cos θ, sin θ), where θ is the angle from the positive x-axis.

What is the difference between sin and cosec?

Cosec is the reciprocal of sin: csc(θ) = 1/sin(θ). While sin gives opposite/hypotenuse, cosec gives hypotenuse/opposite.

Why is tan undefined at 90°?

tan(θ) = sin(θ)/cos(θ). At 90°, cos(90°) = 0, so tan(90°) involves division by zero and is undefined (approaches infinity).

What are the Pythagorean identities?

sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. These follow from the Pythagorean theorem applied to the unit circle.

How is trigonometry used in real life?

In navigation (GPS triangulation), architecture (roof angles), physics (wave equations), music (sound waves), astronomy (distance to stars), and medical imaging (CT scans).

Disclaimer: This calculator provides trigonometric values for educational purposes.

References

  1. Wikipedia. "Trigonometry." en.wikipedia.org
  2. Khan Academy. "Trigonometry." khanacademy.org

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