Trigonometry Chart- Values and Formulas

What Is a Trigonometry Chart?

A trigonometry chart shows the values of sin, cos, and tan for standard angles. It's a quick reference tool that saves you from recalculating ratios every time you need them.

These charts typically cover angles from 0° to 90°, with some extending to 360° for full-circle work. The standard angles appear repeatedly in geometry, physics, and engineering problems.

The Basic Trig Ratios You Must Know

Before memorizing any chart, understand what these ratios actually mean. Memorizing without understanding is a waste of time.

Sine (sin)

Opposite side divided by hypotenuse. In a right triangle, pick an angle, find the side across from it, divide by the longest side.

Cosine (cos)

Adjacent side divided by hypotenuse. Same triangle, pick an angle, find the side next to it (not the hypotenuse), divide by the longest side.

Tangent (tan)

Opposite divided by adjacent. Some people remember this as SOH CAH TOA — Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

The Standard Trigonometry Values Table

Commit these values to memory. They're the foundation for everything else.

That's it. Five angles. Four values each. If you can't recall these instantly, you will struggle with anything more advanced.

How to Read the Chart Quickly

Look at the pattern in the sine column: 0, 1/2, √2/2, √3/2, 1. The values increase as the angle increases from 0° to 90°.

The cosine column mirrors this in reverse: 1, √3/2, √2/2, 1/2, 0. Cosine of 0° equals sine of 90°. They flip.

Tangent starts at 0, climbs toward infinity as you approach 90°. At 90°, tangent is undefined — the line never crosses the y-axis.

Extended Chart: 0° to 360°

Full trigonometry charts include all four quadrants. Here's how the signs work:

• Quadrant I (0° to 90°): All values positive
• Quadrant II (90° to 180°): Sine positive, cosine and tangent negative
• Quadrant III (180° to 270°): Tangent positive, sine and cosine negative
• Quadrant IV (270° to 360°): Cosine positive, sine and tangent negative

This matters when solving problems involving angles beyond 90°. Ignore the signs and you'll get wrong answers every time.

Reciprocal Trig Functions

Three more functions exist. They're less common but show up in calculus and advanced problems.

• Cosecant (csc): 1/sin — the reciprocal of sine
• Secant (sec): 1/cos — the reciprocal of cosine
• Cotangent (cot): 1/tan — the reciprocal of tangent

The values are just inverted. If sin = 1/2, then csc = 2. Simple arithmetic.

Key Trigonometry Formulas

These identities connect the functions and let you simplify complex expressions.

Pythagorean Identity

sin²θ + cos²θ = 1

This is the most useful identity. It works for any angle. Derive tangent from it: tan = sin/cos, so tan²θ + 1 = sec²θ.

Sum and Difference Formulas

• sin(A + B) = sin A cos B + cos A sin B
• sin(A - B) = sin A cos B - cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• cos(A - B) = cos A cos B + sin A sin B

These come up constantly in physics and signal processing. No way around learning them.

Double Angle Formulas

• sin(2θ) = 2 sinθ cosθ
• cos(2θ) = cos²θ - sin²θ

Half-angle formulas exist too, but they're less frequently needed. Start here.

How to Use This Chart in Practice

Download or print the chart. Keep it visible while working problems. Over time, the values stick.

When solving a right triangle problem: identify the known angle, find the corresponding ratio, set up your proportion, solve.

When simplifying an expression: look for opportunities to use Pythagorean identities. Replace sin² + cos² with 1 whenever it appears.

When checking your work: verify that sin² + cos² = 1 holds for your calculated values. If it doesn't, something went wrong.

Common Mistakes to Avoid

• Confusing the ratios. Sin goes with opposite/hypotenuse. Cos goes with adjacent/hypotenuse. Don't mix them up.
• Forgetting the signs. Angles in different quadrants change the sign of every function except sine in QII and cosine in QIV.
• Using degrees when radians are required. Most calculators default to degrees. Check your mode setting before calculating.
• Rounding too early. Keep full values (√3/2) in your calculations. Round only at the final step.

When to Use a Chart vs. a Calculator

A chart gives exact values for standard angles. No decimal approximations, no rounding errors.

A calculator handles non-standard angles: 17°, 42°, 73.5°. Use it when the angle isn't on the chart.

For exams with exact answer requirements, stick to the chart. √3 is exact. 0.866 is an approximation.

Quick Reference: Sine and Cosine Values

The 15° and 75° values are useful for exact calculations involving half-angles. The 22.5° values come from half of 45°.