How to convert between degrees and radians?

1 radian = 180/π degrees ≈ 57.296°, 1 degree = π/180 radians ≈ 0.01745 rad. For example, 45° = π/4 rad ≈ 0.7854 rad.

When is tan undefined?

When the angle is 90°+180°n (n is an integer), cos=0, so tan=sin/cos is undefined. Similarly, when the angle is 0°+180°n, sin=0, so cot is undefined.

What are the trig values for common special angles?

30°: sin=1/2, cos=√3/2, tan=√3/3; 45°: sin=cos=√2/2, tan=1; 60°: sin=√3/2, cos=1/2, tan=√3.

📐 Trigonometric Functions

Calculate sin, cos, tan, cot, sec, csc

Angle (degrees)

Angle Mode

45° = 0.78539816 rad

sin

0.70710678

cos

0.70710678

tan

cot

sec

1.41421356

csc

1.41421356

Common Special Angles

Angle	sin	cos	tan
0°	0	1	0
30°	0.5	0.8660254	0.57735027
45°	0.70710678	0.70710678	1
60°	0.8660254	0.5	1.73205081
90°	1	0	Undefined
120°	0.8660254	-0.5	-1.73205081
135°	0.70710678	-0.70710678	-1
150°	0.5	-0.8660254	-0.57735027
180°	0	-1	0

Trigonometry Calculator Guide

This calculator computes sin, cos, tan, and other trigonometric function values. Enter an angle in degrees or radians to calculate the corresponding trig values, with support for switching between degree and radian modes.

Trigonometric functions are fundamental tools in mathematics and engineering, widely used in surveying, navigation, signal processing, and physics. Always confirm the angle mode (DEG/RAD) before calculating.

Trigonometry Basics

Common trigonometric functions:

- sin(θ) = Opposite ÷ Hypotenuse - cos(θ) = Adjacent ÷ Hypotenuse - tan(θ) = Opposite ÷ Adjacent = sin ÷ cos

Special angle values: - sin 0°=0, sin 30°=0.5, sin 45°=√2/2, sin 60°=√3/2, sin 90°=1 - cos 0°=1, cos 30°=√3/2, cos 45°=√2/2, cos 60°=0.5, cos 90°=0 - tan 0°=0, tan 30°=√3/3, tan 45°=1, tan 60°=√3, tan 90°=undefined

Degree-radian conversion: 1° = π/180 rad

This calculator is for reference only. Please verify results independently.

FAQ

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Trigonometric Functions Calculator Guide

This calculator simultaneously computes six trigonometric function values: sine (sin), cosine (cos), tangent (tan), and their reciprocal functions cotangent (cot), secant (sec), and cosecant (csc). It supports both degree and radian mode switching.

Trigonometric functions bridge angles and side-length ratios, originating from ancient Greek astronomical observations and land surveying. Today they are indispensable in signal processing, wave analysis, architectural structural design, and computer graphics. This tool also provides a quick reference table for common special angles (30°, 45°, 60°, etc.).

Definitions and Identities

【Right Triangle Definition】(Acute angle A)
sin A = Opposite/Hypotenuse, cos A = Adjacent/Hypotenuse
tan A = Opposite/Adjacent, cot A = Adjacent/Opposite
sec A = Hypotenuse/Adjacent, csc A = Hypotenuse/Opposite

【Basic Identities】
sin²θ + cos²θ = 1    (Pythagorean identity)
tan θ = sin θ / cos θ
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

【Common Special Angles】
sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2
cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2

Practical Examples

📋

【Example 1】Find sin(30°) Mode: Degrees (DEG), input 30 Result: sin(30°) = 0.5 Geometric meaning: In a right triangle, the side opposite 30° is half the hypotenuse 【Example 2】Find cos(π/4) Mode: Radians (RAD), input 0.7854 (≈π/4) Result: cos(π/4) ≈ 0.7071 = √2/2 【Example 3】Find tan(60°) Mode: Degrees (DEG), input 60 Result: tan(60°) ≈ 1.732 = √3 Verification: sin(60°)/cos(60°) = (√3/2)/(1/2) = √3 ✓ 【Example 4】Engineering: Slope height Slope angle 30°, slope length 10m → Height h=10×sin(30°)=5m

Important Notes

💡

- Always confirm whether you are in degree (DEG) or radian (RAD) mode before use - tan(90°+180°k) is undefined (cos=0 causes division by zero), displayed as a very large value - cot(0°+180°k) is also undefined (sin=0) - Range of trig functions: sin and cos are in [-1,1], tan and cot can be any real number - Degree to radian: rad = deg × π/180; Radian to degree: deg = rad × 180/π

Application Scenarios

- Architecture: Slope/roof pitch calculation, truss force analysis, antenna elevation angle
- Navigation: GPS triangulation, maritime celestial navigation
- Signal processing: Fourier transform decomposes time-domain signals into sine wave superpositions
- Physics: Simple harmonic motion x=A·sin(ωt+φ), wave interference and diffraction
- Game development: Character movement direction, camera rotation, 3D rotation matrices

Degrees vs Radians In-Depth

Degrees: Divide a full circle into 360 equal parts, each 1°. Originating from the ancient Babylonian sexagesimal system, intuitive and widely used in daily life.

Radians: Measure the central angle by the arc length in units of radius. A full circle = 2π rad ≈ 6.283 rad. The advantage of radians is that they are a dimensionless natural unit — when r=1, arc length equals the radian value.

Why does higher mathematics prefer radians? Because only with radians do the elegant formulas lim(sin x/x)=1 (x→0) and d/dx(sin x)=cos x hold. With degrees, derivatives would include an extra π/180 factor. Therefore, calculus, physics formulas, and programming trig functions all default to radians.

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