What this calculator does
Get the sine, cosine, tangent and the other trig values of any angle, or find an angle from one of those values. Trig functions turn an angle into a ratio, a number comparing two lengths. Picture a circle with a radius of 1 (the unit circle). Put a point on it at your angle, measured counterclockwise from the right-hand side of the horizontal x-axis. The point's left-right position (its x-coordinate) is the cosine. Its up-down position (its y-coordinate) is the sine. The other four functions are built from those two. Tangent is sine divided by cosine. Cosecant, secant and cotangent are 1 divided by sine, cosine and tangent.
This calculator works in both directions. Give it an angle in degrees or radians, and it shows all six values. When the angle is a multiple of 30° or 45°, it also gives the exact form, such as √3/2. Switch to inverse mode and give it a value. You get the angle back from arcsin, arccos or arctan, in degrees and radians.
Where it's used
Why this math matters outside the classroom:
- Tree height. Standing 30 m from a tree and sighting its top at 40°, a hiker estimates its height as 30 × tan 40°, about 25.2 m.
- Ramp angles. A ramp rising 1 m over a 12 m run meets the ground at arctan(1/12), about 4.8°.
- Ladder reach. A 6 m ladder leaning at 75° to the ground reaches 6 × sin 75°, about 5.8 m, up a wall.
- Sound waves. Audio engineers model a 440 Hz tone as sin(2π × 440t) to describe the air pressure at each moment t.
- Solar panels. Installers find that a panel 1.7 m long tilted at 30° rises 1.7 × sin 30° = 0.85 m at the back.
Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.SRT.C.6, HSF.TF.A.2, HSF.TF.A.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose “Trig functions of an angle” to find sin, cos, tan, csc, sec and cot. Or choose “Inverse” to find an angle from a value.
- For an angle, pick the function you want as the main result and type the angle. Set the unit to degrees or radians. Negative angles and angles past 360° are fine.
- For an inverse, pick arcsin, arccos or arctan and type the value. For arcsin and arccos, the value must be between −1 and 1.
- Read the result. All six functions are listed together. Any that would divide by zero at that angle are marked Undefined.
What the inputs mean
- Function
- The function shown as the main result. The other five are always listed below it.
- Angle
- Degrees are the everyday unit, with 360° in a full turn. Radians measure the distance around the edge of a unit circle, with 2π (about 6.2832) in a full turn. Enter radians as a decimal. For example, π/4 is about 0.7853981634.
- Value
- The ratio whose angle you want. A sine or cosine can never be more than 1 or less than −1. So arcsin and arccos won't accept such values, while arctan accepts any number.
Formula and method
sin θ = y, cos θ = x (point on the unit circle) tan θ = sin θ ÷ cos θ csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = cos θ ÷ sin θ radians = degrees × π ÷ 180 arcsin x ∈ [−90°, 90°], arccos x ∈ [0°, 180°], arctan x ∈ (−90°, 90°)
In these formulas, θ (the Greek letter theta) is the angle. The letters x and y are the point's left-right and up-down positions on the unit circle. In the last line, x is the value you give an inverse function, and ∈ means "falls in the range".
The calculator finds sine and cosine first, then divides to get the other four. If the bottom of a division is exactly zero, that ratio is undefined. That happens for tangent and secant at 90° and 270°, and for cosecant and cotangent at 0° and 180°. The calculator shows these as Undefined rather than a huge number.
Computers store π only roughly. So a plain computer calculation of sin 30° gives 0.49999999999999994. To avoid that, the calculator first brings the angle into the range 0° to 360°. It spots the multiples of 30° and 45° and uses their exact values. Radian inputs within a billionth of a degree of one of these angles are treated the same way.
An inverse function has many possible answers, because every ratio repeats each full turn. The calculator gives the standard main answer (the principal value) in the ranges listed above. Textbooks and scientific calculators use the same rule.
Worked example
A 6 m ladder leans against a wall at 70° to the ground. How high up the wall does it reach?
- The height is the side across from the 70° angle, and the ladder is the hypotenuse (the longest side). So height = 6 × sin 70°.
- Enter 70 degrees with sine selected: sin 70° ≈ 0.939693.
- Multiply: 6 × 0.939693 ≈ 5.638 m.
- To go the other way, a 6 m ladder reaching 5.638 m up makes an angle of arcsin(5.638 ÷ 6) with the ground. That's back to about 70°.
Result: The ladder reaches about 5.638 m up the wall. The calculator also lists cos 70° ≈ 0.34202. That gives the distance from the wall: 6 × 0.34202 ≈ 2.052 m.
Understanding the result
Sine and cosine are always between −1 and 1. Tangent, cotangent, secant and cosecant can be any size. They grow without limit near the angles where they're undefined.
Whether each value is positive or negative depends on the quadrant, the quarter of the circle the angle points into. Sine is positive above the x-axis (0° to 180°). Cosine is positive to the right of the y-axis (between 270° and 90°). Tangent is positive where sine and cosine have the same sign.
An exact value is shown only when one exists as a simple square-root form. Other angles give decimals rounded to the precision you chose.
Assumptions and limitations
- Angles are measured counterclockwise from the positive x-axis, the standard math convention.
- Inverse results are principal values, the standard main answers.
- Inverse mode covers arcsin, arccos and arctan. For arccsc, arcsec or arccot, first take 1 divided by your value: arccsc x = arcsin(1 ÷ x).
- Exact forms are given only for multiples of 30° and 45°.
- Grads (400 to a full turn) aren't supported. Multiply by 0.9 to change them to degrees.
Frequently asked questions
How do I remember which ratio is which?
In a right triangle, sine is opposite ÷ hypotenuse, cosine is adjacent ÷ hypotenuse and tangent is opposite ÷ adjacent. Many people remember this as SOH CAH TOA. The unit-circle definitions used here match these for angles between 0° and 90°. They also stretch them to work for every angle.
Why is tan 90° undefined?
Tangent is sine divided by cosine, and cos 90° is exactly 0. Dividing by zero has no answer, so tan 90° doesn't exist. As the angle gets closer to 90° from below, the tangent grows without limit.
Why does arcsin(2) give an error?
Sine is the height of a point on a circle of radius 1. So it can never be more than 1 or less than −1. No angle has a sine of 2, so arcsin(2) has no real answer. The same goes for arccos.
What is the difference between degrees and radians?
Both measure angles. A full turn is 360° or 2π radians. So 180° equals π radians, and 1 radian is about 57.2958°. Radians are standard in calculus and most programming languages. Degrees are more common in everyday life.
Is arcsin the same as 1 ÷ sin?
No. Arcsin (also written sin⁻¹) undoes the sine: it takes a ratio and gives back an angle. The value 1 ÷ sin θ is a different function, called the cosecant.
How this calculator is checked
Method: Unit-circle definitions of the six trigonometric functions; principal-value conventions for the inverse functions; exact values at multiples of 30° and 45°. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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