OneTaskBy Tim King · Updated
Trig Calculator
Degrees or Radians, Exact Values for Special Angles, Plus Inverse Functions
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How the Trig Calculator works
This trig calculator gives the sine, cosine, tangent, cosecant, secant and cotangent of any angle, in degrees or radians. For multiples of 15° it shows the exact value, such as √3/2, beside the decimal. Switch to arcsin, arccos or arctan to go the other way and find the angle from a ratio.
Picture a point moving round a circle of radius 1, starting on the positive x-axis. After it has turned through an angle θ, its x-coordinate is cos θ and its y-coordinate is sin θ. Tangent is the ratio of the two, sin θ ÷ cos θ, and the other three functions are reciprocals: csc = 1/sin, sec = 1/cos and cot = 1/tan.
How the trig calculator works
The calculator converts the angle to degrees, finds where it sits on the circle and works out its reference angle, the acute angle it makes with the x-axis. The formula for every function is the value at the reference angle with a sign set by the quadrant. Where a function divides by zero, such as tan 90°, the answer is undefined. For the inverses the answer is the principal value, and the calculator also lists the second angle in 0° to 360° with the same sine or cosine.
First example. sin 30° = 1/2 exactly. cos 30° = √3/2 ≈ 0.8660 and tan 30° = √3/3 ≈ 0.5774.
Second example. cos 225° sits in quadrant III with a reference angle of 45°. Cosine is negative there, so cos 225° = −√2/2 ≈ −0.7071.
How many degrees is arcsin 0.5? 30°, or π/6 in radians. 150° has the same sine, so both angles solve sin θ = 0.5 between 0° and 360°.
Exact sin, cos and tan values from 0° to 360° in 15° steps
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 15° | π/12 | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 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 |
| 75° | 5π/12 | (√6 + √2)/4 | (√6 − √2)/4 | 2 + √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 105° | 7π/12 | (√6 + √2)/4 | −(√6 − √2)/4 | −(2 + √3) |
| 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 |
| 165° | 11π/12 | (√6 − √2)/4 | −(√6 + √2)/4 | −(2 − √3) |
| 180° | π | 0 | −1 | 0 |
| 195° | 13π/12 | −(√6 − √2)/4 | −(√6 + √2)/4 | 2 − √3 |
| 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 |
| 255° | 17π/12 | −(√6 + √2)/4 | −(√6 − √2)/4 | 2 + √3 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 285° | 19π/12 | −(√6 + √2)/4 | (√6 − √2)/4 | −(2 + √3) |
| 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 |
| 345° | 23π/12 | −(√6 − √2)/4 | (√6 + √2)/4 | −(2 − √3) |
| 360° | 2π | 0 | 1 | 0 |
Tips
- Check the unit. sin 30 in radians is about −0.988, not 0.5. Most wrong answers come from a calculator left in the other mode.
- Use the reference angle. Learn the values at 30°, 45° and 60° and you can find every multiple of 15° by sign alone.
- Two angles, one sine. Inverse functions return one angle. Add the second from the result list when solving an equation.
FAQ
What is the difference between degrees and radians?
Both measure the size of an angle. A full turn is 360° or 2π radians, so 1 radian is about 57.3°. Radians come from the circle itself: an angle of 1 radian cuts off an arc as long as the radius. Calculus formulas such as the derivative of sin x = cos x only hold in radians.
Why is tan 90° undefined?
Tangent is sin ÷ cos, and cos 90° = 0. Dividing by zero has no answer. On a graph, tan x shoots up to infinity just before 90° and comes back from minus infinity just after, so there is no single value at 90° itself. The same happens at 270° and every 180° from there.
What does ASTC or All Students Take Calculus mean?
It is a memory aid for which functions are positive in each quadrant. In quadrant I all are positive, in II only sine, in III only tangent and in IV only cosine. The calculator applies these signs for you, and the Position line tells you which quadrant your angle is in.
How do you find the exact value of sin 15°?
Use the difference formula: sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°. That is (√2/2)(√3/2) − (√2/2)(1/2), which simplifies to (√6 − √2)/4 ≈ 0.2588. The table above lists the exact values for every 15° step.
See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.
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