Trigonometry Calculator

The trigonometric functions sine, cosine and tangent link an angle to the coordinates of a point on the unit circle. This trigonometry calculator gives all six values for any angle in degrees or radians, with exact answers such as √3/2 for special angles, plus the reference angle and quadrant explained.

Try:

How to use the trigonometry calculator

  1. Enter the angle, such as 150 or pi/6.
  2. Choose degrees or radians.
  3. Press Calculate to see all six trig values and the unit circle.

Formula

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}

Worked example: θ = 150°

  • Angle θ: 150
  • Unit: Degrees

✓ Answer checked

sin⁡θ=12,cos⁡θ=−32,tan⁡θ=−33\sin\theta = \frac{1}{2},\quad \cos\theta = -\frac{\sqrt{3}}{2},\quad \tan\theta = -\frac{\sqrt{3}}{3}
csc θ
22
sec θ
−233-\frac{2\sqrt{3}}{3}
cot θ
−3-\sqrt{3}
Decimals
sin⁡≈0.5,  cos⁡≈−0.8660254038,  tan⁡≈−0.5773502692\sin \approx 0.5,\; \cos \approx -0.8660254038,\; \tan \approx -0.5773502692
Quadrant
2
(cos θ, sin θ) = (-0.866, 0.500) 11
On the unit circle, the point at angle θ = 150° has coordinates (cos θ, sin θ).
Step-by-step working (3 steps)

  1. Convert the angle π rad = 180°

    Multiply degrees by π/180 to get radians.

    θ=150∘=2.617993878 rad\theta = 150^\circ = 2.617993878\text{ rad}

  2. Find the reference angle Reference angle

    The angle is in quadrant 2. The reference angle is the acute angle to the x-axis.

    θref=30∘\theta_{\text{ref}} = 30^\circ

  3. Use exact values and signs All Students Take Calculus

    In quadrant 2, only sin is positive (ASTC rule).

    sin⁡θ=12,  cos⁡θ=−32,  tan⁡θ=−33\sin\theta = \frac{1}{2},\; \cos\theta = -\frac{\sqrt{3}}{2},\; \tan\theta = -\frac{\sqrt{3}}{3}
Formulas used
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
How this was checked
  • ✓ sin²θ + cos²θ = 1 holds.
  • ✓ The exact values agree with the decimal values.

Frequently asked questions

What is sin 30°?

Exactly 1/2. The special angles 30°, 45° and 60° all have exact values involving √2 or √3.

Why is tan 90° undefined?

tan θ = sin θ ÷ cos θ, and cos 90° = 0. Dividing by zero is undefined.

How do I know the sign in each quadrant?

Use ASTC: in quadrant 1 All are positive, in 2 only Sine, in 3 only Tangent, and in 4 only Cosine.

Related calculators

Degrees to Radians Calculator

Convert degrees to radians as an exact multiple of π (like 135° = 3π/4) and as a decimal, with the steps and a unit circle diagram.

Radians to Degrees Calculator

Convert radians to degrees, including inputs like 5pi/6, with degrees-minutes-seconds and a unit circle diagram.

Triangle Calculator

Solve any triangle from three known parts: sides, angles, area, perimeter and type, using the laws of sines and cosines.

Pythagorean Theorem Calculator

Find the hypotenuse or a missing side of a right triangle using a² + b² = c².

Related guides