Math
Trigonometric Functions Calculator
Enter any angle and get all six trigonometric functions at once. mathcheck shows exact values such as √3/2 or 2 − √3 for special angles, decimals for everything else, flags undefined results like tan 90°, and draws the angle on the unit circle with its quadrant, reference angle, and coterminal angles.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
sin θ
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- cos θ
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- tan θ
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- csc θ
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- sec θ
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- cot θ
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- Quadrant
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More details (2 more)
- Reference angle
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- Radians
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Study path
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What this calculator solves
Given one angle, it evaluates sine, cosine, tangent, cosecant, secant, and cotangent. The angle can be typed in degrees (150 or 30° 15′), An angle unit where a full turn is 2π instead of 360 degrees. (π/6, 5pi/6, or 0.5236), multiples of π, gradians, or turns. Negative angles and angles larger than 360° are reduced to a coterminal angle first.
Formulas
Every value is computed from the The small positive angle between a given angle's arm and the x-axis. and then given the sign of its One of the four regions the x- and y-axes cut the coordinate plane into.. The mnemonic ASTC (All, Sine, Tangent, Cosine) lists which functions are positive in Quadrants I through IV.
Exact values for special angles
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 15° (π/12) | (√6 − √2)/4 | (√6 + √2)/4 | 2 − √3 |
| 18° (π/10) | (√5 − 1)/4 | √(10 + 2√5)/4 | √(25 − 10√5)/5 |
| 30° (π/6) | 1/2 | √3/2 | √3/3 |
| 36° (π/5) | √(10 − 2√5)/4 | (1 + √5)/4 | √(5 − 2√5) |
| 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 |
How to use it
- Choose the angle unit. Radians accept π fractions and decimals; degrees accept decimals, fractions like 180/7, and degrees-minutes-seconds.
- Type the angle. Negative angles and angles past one full turn are fine.
- Read sin, cos, and tan in the header, all six functions in the grid, and the quadrant, reference angle, and coterminal angles below.
- Open Show the work to see the conversion, the reference angle, the sign chart, and each function built from the reference angle.
How to read the answer
- A value shown as 1/2 (0.5) is exact on the left and rounded on the right.
- undefined means the function divides by zero at that angle: tan and sec are undefined at 90° and 270°, csc and cot at 0° and 180°.
- sin and cos always stay between -1 and 1; csc and sec are never strictly between -1 and 1.
- The unit-circle point is (cos θ, sin θ), so cos is the horizontal leg and sin is the vertical leg of the right triangle in the figure.
Common mistakes and edge cases
- Mixing units: sin(30) in radian mode is -0.988, not 0.5. Check the unit selector before trusting a result.
- Forgetting quadrant signs: sin 150° is positive because 150° is in Quadrant II, but cos 150° is negative.
- Using a rounded reference angle: keep the exact angle until the last step so radical forms survive.
- Reading 1/√2 and √2/2 as different values: they are equal; this calculator shows the rationalized form √2/2.
Worked examples
Quadrant II special angle
θ = 150°
sin θ
1/2 (0.5)
Radians with π
θ = 5π/6
sin θ
1/2 (0.5)
Negative angle
θ = -45°
sin θ
-√2/2 (-0.707107)
More than one turn
θ = 750° (coterminal with 30°)
sin θ
1/2 (0.5)
Undefined tangent
θ = 90°
sin θ
1
18° family
θ = 18° has a golden-ratio sine
sin θ
(√5 − 1)/4 (0.309017)
Decimal radians
θ = 0.5236 rad (close to π/6)
sin θ
0.500001
Gradians
θ = 100 grad = 90°
sin θ
1
Unreadable input
θ = 'thirty'
sin θ
—
Frequently asked questions
Why is tan 90° undefined?+
tan θ = sin θ / cos θ, and cos 90° = 0. Division by zero has no value, so tan 90° (and sec 90°) are undefined. The same happens at 270°, and csc and cot are undefined at 0° and 180°.
How do I enter radians?+
Select Radians and type π/6, pi/6, 5π/6, 2pi/3, or a decimal like 0.5236. If you type an expression with π while Degrees is selected, the calculator reads it as radians and tells you.
What is a reference angle?+
The acute angle between the terminal side and the x-axis. Every trig value of the original angle equals the same function of the reference angle, up to sign. For 150° the reference angle is 30°, so sin 150° = sin 30° = 1/2.
Which functions are positive in each quadrant?+
Use ASTC: All are positive in Quadrant I, Sine (and csc) in Quadrant II, Tangent (and cot) in Quadrant III, Cosine (and sec) in Quadrant IV.
Why does 0.5236 rad not give exactly 1/2?+
0.5236 is a rounded version of π/6 = 0.52359877... The calculator uses exactly what you typed, so it returns 0.499999 and points out that π/6 is nearby. Type π/6 to get 1/2.
What are gradians and turns?+
A gradian is 1/400 of a full circle (100 grad = 90°). A turn is one full revolution (0.25 turn = 90°). Both are converted to degrees before the functions are evaluated.
Does the calculator handle angles bigger than 360° or negative angles?+
Yes. It subtracts or adds full turns to find the coterminal angle between 0° and 360°, then evaluates from there. sin 750° = sin 30° = 1/2, and sin(-45°) = -√2/2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026