Trigonometry Calculator

Quick Answer

Last updated: 10 November 2026 ·How we calculate this: the unit circle definitions of sin, cos and tan

How This Trigonometry Calculator Works

Choose Functions of an angle, enter an angle and pick degrees or radians. The calculator gives all six trig functions, converts the angle to the other unit and shows exact values for common angles such as 30°, 45° and 60°. Choose Angle from a value to go backwards: enter a value and pick arcsin, arccos or arctan to get the angle.

How to Use the Trigonometry Calculator

1. Choose whether you want the functions of an angle or an angle from a value.
2. Enter the angle and choose degrees or radians, or enter the value.
3. For an inverse, pick arcsin, arccos or arctan.
4. Press Calculate.
5. Read the results and the exact values for special angles.

Trigonometry Formulas

• Unit circle: the point at angle t is (cos t, sin t)
• Tangent: tan t = sin t ÷ cos t
• Reciprocals: csc t = 1 ÷ sin t, sec t = 1 ÷ cos t, cot t = 1 ÷ tan t
• Pythagorean identity: sin² t + cos² t = 1
• Degrees to radians: radians = degrees × π ÷ 180

Key Terms

• Unit circle: a circle of radius 1 centered at the origin, used to define the trig functions.
• Hypotenuse: the longest side of a right triangle, opposite the right angle.
• Reference angle: the acute angle between the terminal side of an angle and the x-axis.
• Radian: the angle that cuts off an arc equal to the radius; 2π radians make a full turn.
• Inverse function: a function that undoes another, such as arcsin undoing sin.

Worked Example

Find the trig values of 135°.

• 135° is in the second quadrant, with a reference angle of 45°.
• sin 135° = +√2/2 ≈ 0.707107 (sine is positive in quadrant II)
• cos 135° = −√2/2 ≈ -0.707107 (cosine is negative in quadrant II)
• tan 135° = sin ÷ cos = -1

Real-Life Examples

• Ladder height: a 6 m ladder at 70° to the ground reaches 6 × sin 70° = 5.64 m up the wall.
• Ramp angle: a ramp rising 1 m over 12 m has an angle of arctan(1/12) = 4.76°.
• Tree height: standing 20 m away and looking up 35° to the top, the tree is 20 × tan 35° = 14.00 m above eye level.
• Roof slope: a 6 in 12 roof pitch is arctan(0.5) = 26.57°.

Degrees vs Radians

• Degrees split a full turn into 360 parts. They are common in everyday use and construction.
• Radians measure the arc length on a unit circle: a full turn is 2π ≈ 6.283 radians. They are standard in calculus and physics.
• Converting: 180° = π radians, so multiply degrees by π/180 to get radians.

Exact Trig Values

Signs in Each Quadrant

A handy way to remember the signs is “All Students Take Calculus”, going anticlockwise from the first quadrant:

• Quadrant I (0° to 90°): all functions are positive.
• Quadrant II (90° to 180°): only sin and csc are positive.
• Quadrant III (180° to 270°): only tan and cot are positive.
• Quadrant IV (270° to 360°): only cos and sec are positive.

Inverse Trig Functions

The inverse functions answer “which angle has this value?”. Because sin, cos and tan repeat, each inverse gives one answer in a set range: arcsin gives −90° to 90°, arccos gives 0° to 180°, and arctan gives an angle between −90° and 90°. For example, arcsin(0.5) = 30°, although 150° also has a sine of 0.5. Add or subtract 360° to find other solutions.

Common Mistakes to Avoid

• Using the wrong angle unit. sin 30 in radians is about −0.988, not 0.5.
• Expecting a value for tan 90°. cos 90° is 0, so tan is undefined.
• Reading sin⁻¹ as 1 ÷ sin. sin⁻¹ means arcsin; 1 ÷ sin is csc.
• Forgetting other solutions. arcsin gives one angle, but there are often two in a full turn.
• Entering arcsin or arccos of a number above 1. They only accept −1 to 1.

Frequently Asked Questions

References & Sources