In algebra, precalculus and calculus, angles usually appear in radians written as fractions of π: π/6, 3π/4, 11π/6. Typing these into an ordinary calculator turns them into long decimals and loses the exact answers your teacher expects. This calculator keeps the angle exact from start to finish. Enter the coefficient of π and it returns sine, cosine, tangent, cosecant, secant and cotangent as radicals such as √3/2 or 2 − √3, with decimals alongside.
How to use the trig functions of π calculator
- In Angle as a multiple of π, type the number that multiplies π:
5/6for 5π/6,-1/4for −π/4,2for 2π. Mixed numbers (1 1/2), decimals (0.25) and forms like5π/6are accepted. - Choose a headline function to show one value in large type. All six values are always listed.
- Read the exact and decimal values, the angle in degrees, the coterminal angle between 0 and 2π, the quadrant and the reference angle.
- Open Show the work for the reasoning, and scan the table of the whole π/q family to see where your angle sits.
Formulas for angles in terms of π
An angle of (p/q)·π radians converts to degrees by replacing π with 180°:
Adding or subtracting 2π gives a coterminal angle with the same trig values, and the reference angle depends on the quadrant:
The sine and cosine of the reference angle, with the quadrant’s signs, give every other value: tan θ = sin θ ÷ cos θ, csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ and cot θ = cos θ ÷ sin θ.
Worked example: θ = 29π/6
1. Convert to degrees. 29π/6 × 180°/π = 29 × 30° = 870°.
2. Remove full turns. 29π/6 − 4π = 29π/6 − 24π/6 = 5π/6 (150°).
3. Find the quadrant and reference angle. 5π/6 lies between π/2 and π, so it is in Quadrant II, and the reference angle is π − 5π/6 = π/6.
4. Apply the signs. In Quadrant II sine is positive and cosine is negative: sin(29π/6) = 1/2 and cos(29π/6) = −√3/2 ≈ −0.8660254038.
5. Finish. tan(29π/6) = −√3/3, csc = 2, sec = −2√3/3 and cot = −√3.
Which fractions of π have exact values?
The denominator q decides whether a neat answer exists:
| Denominator | Angle step | Typical exact values |
|---|---|---|
| 1, 2 | 180°, 90° | 0, ±1 (tangent may be undefined) |
| 3, 6 | 60°, 30° | 1/2, √3/2, √3/3, √3 |
| 4 | 45° | √2/2, 1 |
| 12 | 15° | (√6 ± √2)/4, 2 ± √3 |
| 8 | 22.5° | √(2 ± √2)/2, √2 ± 1 |
| 5, 10 | 36°, 18° | (√5 ± 1)/4 and related forms |
Angles such as π/7 and π/9 have no expression using real square roots. That follows from the Gauss–Wantzel theorem on which regular polygons can be built with a straightedge and compass: a regular 14-gon or 18-gon cannot, so cos(π/7) and cos(π/9) cannot be written with square roots alone. A few other denominators, such as 15, do have square-root forms, but they are long and rarely used, so the calculator lists those as decimals.
Common mistakes with π
- Reading π/6 as 1/6. The fraction multiplies π. π/6 is 30°, about 0.5236 radians, not 0.1667.
- Mixing up sin(π)/6 and sin(π/6). The first is 0 because sin π = 0; the second is 1/2. On a handheld calculator, close the parentheses after the 6.
- Forgetting full turns. 13π/6 is not a new angle; it is π/6 plus one full turn, so its values match π/6 exactly.
- Dropping a negative sign. −π/4 lies in Quadrant IV, so sin(−π/4) = −√2/2 while cos(−π/4) = +√2/2. Cosine is an even function and sine is odd.
For angles in degrees or decimals, use the trigonometric functions calculator; to see every standard angle placed on the circle at once, open the unit circle calculator.
Frequently asked questions
How do I enter 5π/6?
Type 5/6 in the box; the calculator multiplies it by π for you. You can also type 5π/6 or 5pi/6 directly. Negative angles such as -3/4 and mixed numbers such as 1 1/2 (meaning 3π/2) work too.
How do I convert a multiple of π to degrees?
Replace π with 180°. For example, 7π/4 = 7 × 180° ÷ 4 = 315°, and π/12 = 15°.
Why do some angles only show decimals?
Exact square-root forms exist only for certain fractions of π. The calculator knows all multiples of π/12, π/10 and π/8. An angle like π/7 has no expression using real square roots, so it is shown to 10 decimal places.
What does the table under the result show?
For denominators 2, 3, 4, 5, 6, 8, 10 and 12 it lists every multiple of π/q from 0 to 2π with exact sine, cosine and tangent, and highlights your angle. It is a quick way to see the pattern around the whole circle.