Trigonometric Functions Calculator

Calculate all six trigonometric functions: sin, cos, tan, sec, csc, and cot

Calculate Trigonometric Functions

Angle Information

Quadrant:
I
Degrees:
45°
Radians:
0.7854 rad
Function Signs:
sin(+) cos(+) tan(+)

Trigonometric Function Values

Primary Functions

sin(θ)
0.707107
Exact: √2/2
cos(θ)
0.707107
Exact: √2/2
tan(θ)
1
Exact: 1

Reciprocal Functions

sec(θ) = 1/cos(θ)
1.414214
Exact: √2
csc(θ) = 1/sin(θ)
1.414214
Exact: √2
cot(θ) = 1/tan(θ)
1
Exact: 1

Alternative Forms

cot(θ) = cos(θ)/sin(θ)
0.707107/0.707107
sin²(θ) + cos²(θ)
1
1 + tan²(θ)
2

Common Angle Examples

SOHCAHTOA Mnemonic

SOH - Sine

sin(θ) = Opposite / Hypotenuse

Sine equals Opposite over Hypotenuse

CAH - Cosine

cos(θ) = Adjacent / Hypotenuse

Cosine equals Adjacent over Hypotenuse

TOA - Tangent

tan(θ) = Opposite / Adjacent

Tangent equals Opposite over Adjacent

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Reciprocal Trigonometric Functions

Secant (sec)

sec(θ) = 1/cos(θ) = Hypotenuse/Adjacent

Secant is the reciprocal of cosine. It's undefined when cos(θ) = 0 (at 90°, 270°, etc.)

Cosecant (csc)

csc(θ) = 1/sin(θ) = Hypotenuse/Opposite

Cosecant is the reciprocal of sine. It's undefined when sin(θ) = 0 (at 0°, 180°, 360°, etc.)

Cotangent (cot)

cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) = Adjacent/Opposite

Cotangent is the reciprocal of tangent. It's undefined when tan(θ) = 0 (at 0°, 180°, 360°, etc.)

Unit Circle Reference

Quadrant I (0° to 90°)

All functions positive

Quadrant II (90° to 180°)

Only sin and csc positive

Quadrant III (180° to 270°)

Only tan and cot positive

Quadrant IV (270° to 360°)

Only cos and sec positive

Special Angles

Angle
sin
cos
tan
0
1
0
30°
1/2
√3/2
√3/3
45°
√2/2
√2/2
1
60°
√3/2
1/2
√3
90°
1
0

Quick Tips

Remember SOHCAHTOA for the basic three functions

Reciprocal functions: sec=1/cos, csc=1/sin, cot=1/tan

Use the unit circle to remember special angle values

Function signs depend on the quadrant

Pythagorean identity: sin²θ + cos²θ = 1

Understanding Trigonometric Functions

What are Trigonometric Functions?

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in mathematics, physics, engineering, and many other fields.

The Six Functions

  • Primary: sine (sin), cosine (cos), tangent (tan)
  • Reciprocal: secant (sec), cosecant (csc), cotangent (cot)

Applications

Engineering

Wave analysis, signal processing, structural analysis

Physics

Oscillations, waves, circular motion, electromagnetic fields

Computer Graphics

3D transformations, rotations, animations

Fundamental Identities

Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Reciprocal Identities

sec θ = 1/cos θ
csc θ = 1/sin θ
cot θ = 1/tan θ