Why is tan(90°) undefined?

On the unit circle, 90° corresponds to the point (0, 1). Tangent is defined as tan(θ) = sin(θ) / cos(θ). At 90°, cos(90°) = 0, so you would be dividing by zero, which is undefined. Mathematically, the tangent function has a vertical asymptote at 90°. For a deeper explanation, see resources such as Khan Academy trigonometry review .

How can I quickly memorize common trigonometric values?

For the special angles 0°, 30°, 45°, 60°, and 90°, you can remember the sine values as √0/2, √1/2, √2/2, √3/2, √4/2, and then use symmetry on the unit circle to get cosine. The CAST rule (All, Sin, Tan, Cos) helps you remember which functions are positive in each quadrant. Our Memory Tips section summarizes these patterns so you do not need to re-derive them every time.

Is radian measure more important than degrees?

In advanced mathematics, calculus, and physics, radians are usually preferred because trigonometric identities and derivatives look simpler in radians (for example, the derivative of sin(x) is cos(x) only when x is in radians). However, degrees are still useful for geometry and everyday angle measurements. A good strategy is to be comfortable in both systems and use degree–radian conversion formulas fluently.

How are sin, cos, and tan related on the unit circle?

On the unit circle, each angle θ corresponds to a point (x, y) = (cos(θ), sin(θ)). Tangent is the ratio tan(θ) = sin(θ) / cos(θ), which can be interpreted as the slope of the line from the origin to that point. The Pythagorean identity sin²(θ) + cos²(θ) = 1 comes directly from the fact that the unit circle has radius 1.

How does the unit circle explain trigonometric signs in each quadrant?

In Quadrant I, both x and y are positive, so sin, cos, and tan are all positive. In Quadrant II, x is negative and y is positive, so sin is positive while cos and tan are negative. In Quadrant III, both x and y are negative, so tan is positive while sin and cos are negative. In Quadrant IV, x is positive and y is negative, so cos is positive while sin and tan are negative. Our Quadrant Sign Summary table visualizes these patterns at a glance.

📊 Trigonometry Table (Complete)

Trigonometry table (0°–360°) with sin, cos, tan in degrees and radians, special-angle exact values, quadrant signs, and any-angle lookup.

Last updated: 2025-10-21 — Compiled and reviewed by Calvin (Math Research, FreeCalculators.app)

A flagship trigonometry reference page with sin, cos, and tan values from 0° to 360°. This guide combines special-angle exact values, degree and radian tables, quadrant sign rules, degree–radian conversion formulas, memory tips, and a full per-degree lookup table to support learning, exams, engineering work, and physics problems.

Trigonometry Table (Degrees)

Complete table showing sin, cos, and tan values for special angles in degrees.

Angle (°)	Angle (rad)	sin(θ)	cos(θ)	tan(θ)
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	Undefined (∞)
120°	2π/3	√3/2	-1/2	-√3
135°	3π/4	√2/2	-√2/2	-1
150°	5π/6	1/2	-√3/2	-√3/3
180°	π	0	-1	0
210°	7π/6	1/2	-√3/2	√3/3
225°	5π/4	√2/2	-√2/2	1
240°	4π/3	√3/2	-1/2	√3
270°	3π/2	-1	0	Undefined (∞)
300°	5π/3	-√3/2	1/2	-√3
315°	7π/4	-√2/2	√2/2	-1
330°	11π/6	-1/2	√3/2	-√3/3
360°	2π	0	1	0

Complete Trigonometry Table (0°–360°)

Complete lookup table showing sin, cos, and tan values for every degree from 0° to 360°.

Quadrant Sign Diagram

Summary of trigonometric function signs in each quadrant of the unit circle.

I: All trigonometric values are positive in Quadrant I.

II: sin is positive, cos and tan are negative in Quadrant II.

III: tan is positive, sin and cos are negative in Quadrant III.

IV: cos is positive, sin and tan are negative in Quadrant IV.

Degree ↔ Radian Conversion

Formulas and examples for converting between degrees and radians.

To Radian

rad = degree × π / 180

Multiply the degree value by π and divide by 180.

To Degree

degree = rad × 180 / π

Multiply the radian value by 180 and divide by π.

Examples

30° = π/6 rad

45° = π/4 rad

60° = π/3 rad

Memory Tips

Helpful techniques to memorize trigonometric values.

For 0°, 30°, 45°, 60°, 90°: sin values are √0/2, √1/2, √2/2, √3/2, √4/2 (in order).

For 45°: sin = cos = √2/2, tan = 1.

30° and 60° are symmetric: sin(30°) = cos(60°) = 1/2, sin(60°) = cos(30°) = √3/2.

Unit circle five-point method: memorize values at 0°, 90°, 180°, 270°, 360° first.

Use the CAST rule: All, Sin, Tan, Cos (positive in quadrants I, II, III, IV respectively).

Trigonometric Function Properties

Summary of key properties for sin, cos, and tan functions.

Function	Period	Parity	Domain	Range
sin(x)	2π	Odd	All real numbers	[-1, 1]
cos(x)	2π	Even	All real numbers	[-1, 1]
tan(x)	π	Odd	All real numbers except π/2 + kπ	All real numbers

Applications

Common use cases for trigonometry tables.

Homework Lookup
Trigonometry Exams
Engineering Calculations
Physics Rotation Problems
Unit Circle Coordinate Understanding

Frequently Asked Questions

Key questions students and engineers often ask when using trigonometry tables.

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