Inverse Trig Calculator — arcsin, arccos, arctan

An inverse trig calculator turns a ratio back into an angle. Since sin 60° = √3/2, arcsin(√3/2) = 60°, which is π/3 radians. The answer always comes from the function’s principal range, here −90° to 90°. Every angle with sin θ = √3/2 is 60° + 360°k or 120° + 360°k.

Embed this calculator on your site

Free to use on class pages, blogs and course pages. Copy the code below and paste it into your page:

<iframe src="https://www.effortlessmath.com/math-topics/inverse-trig-calculator/?em_embed=1" width="100%" height="720" style="border:0;max-width:100%" title="Inverse Trig Calculator — arcsin, arccos, arctan by Effortless Math" loading="lazy"></iframe><p><a href="https://www.effortlessmath.com/math-topics/inverse-trig-calculator/">Inverse Trig Calculator — arcsin, arccos, arctan</a> by <a href="https://www.effortlessmath.com/">Effortless Math</a></p>Copy code

How to use the inverse trig calculator

1. Pick a tab. One function handles arcsin, arccos, arctan, arccsc, arcsec and arccot. Right triangle finds an angle from two sides. All six at once lists every inverse of one number.
2. Type the value as a decimal (0.6), a fraction (−1/2) or with a root (√3/2, sqrt(2)/2, 1/√3). Roots and simple fractions give exact answers such as π/6.
3. Choose degrees or radians for the main answer and how many decimal places you want. Both units are always shown.
4. Read the principal value, every solution in [0°, 360°), the general solution and the unit circle picture. Open Show steps for the full working.

What inverse trig functions do

A trig function takes an angle and gives a ratio: sin 30° = 1/2. An inverse trig function goes the other way. It takes the ratio and returns the angle: arcsin(1/2) = 30°. The notation sin⁻¹(x) means the same thing as arcsin(x). It does not mean 1/sin x, which is csc x.

There is a catch. Infinitely many angles share the same sine (30°, 150°, 390° and so on), so the inverse would not be a function. To fix this, each inverse returns one angle from a fixed interval called the principal range.

Principal ranges of the inverse trig functions

The ranges are chosen so that each function covers every possible output exactly once. arcsin and arctan use Quadrants I and IV, so negative inputs give negative angles. arccos and arccot use Quadrants I and II, so negative inputs give obtuse angles. Some software uses (−90°, 90°] for arccot, so check which convention your course uses.

The reciprocal functions reduce to the main three: arccsc x = arcsin(1/x) and arcsec x = arccos(1/x). For positive x, arccot x = arctan(1/x). For negative x, arccot x = 180° + arctan(1/x).

Exact values of arcsin, arccos and arctan

These come from the 30°-60°-90° and 45°-45°-90° triangles. If your value is in this table, the calculator shows the exact angle.

For arctan: arctan(√3/3) = 30°, arctan(1) = 45°, arctan(√3) = 60°, and the negatives give −30°, −45° and −60°. Note that √3/3 = 1/√3 and √2/2 = 1/√2.

Finding all solutions, not just the principal value

A calculator’s sin⁻¹ key gives only the principal value. To solve an equation such as sin θ = 0.6 on [0°, 360°), use symmetry:

• sin and csc: the second angle is 180° − θ.
• cos and sec: the second angle is 360° − θ.
• tan and cot: the second angle is θ + 180°, because these repeat every 180°.

Add 360°k (or 2πk) to each angle for the general solution. For tan and cot, θ + 180°k covers both angles in one line.

Worked examples

Example 1: arcsin(√3/2)

√3/2 is in the exact-value table: sin 60° = √3/2, and 60° lies in [−90°, 90°]. So arcsin(√3/2) = 60° = π/3. The other angle on [0°, 360°) is 180° − 60° = 120°, so sin θ = √3/2 when θ = 60° + 360°k or θ = 120° + 360°k.

Example 2: arccos(−0.8)

−0.8 is not a special value, so use a calculator in degree mode: arccos(−0.8) ≈ 143.1301° ≈ 2.4981 rad. The answer is obtuse, as arccos of a negative number always is. The other solution of cos θ = −0.8 is 360° − 143.1301° = 216.8699°.

Example 3: arccot(−√3)

cot θ = −√3 means tan θ = −1/√3, and arctan(−1/√3) = −30°. arccot must land in (0°, 180°), so add 180°: arccot(−√3) = 150° = 5π/6. Check: cot 150° = cos 150°/sin 150° = (−√3/2)/(1/2) = −√3.

Common mistakes with inverse trig

• Reading sin⁻¹ x as 1/sin x. The −1 means inverse function, not a power.
• Wrong calculator mode. arcsin(0.5) is 30 in degree mode but 0.5236 in radian mode. Both are correct, in different units.
• Stopping at one answer. “Solve sin θ = 0.6 for 0° ≤ θ < 360°” has two answers, 36.87° and 143.13°.
• Taking arcsin of a number outside [−1, 1]. arcsin(2) does not exist, because sine never exceeds 1. The calculator explains why instead of showing an error code.
• Using arctan(1/x) for arccot of a negative x. That gives a negative angle outside (0°, 180°). Add 180°.

Where inverse trig is used

Inverse trig finds angles from measurements: the angle of a ramp from its rise and run (arctan), the angle of a ladder from its height and length (arcsin), the direction of a force from its components, or a bearing from east and north distances. In calculus, arctan and arcsin appear as antiderivatives, for example ∫ 1/(1 + x²) dx = arctan x + C.

Inverse Trig practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you do inverse trig on a calculator?

Press SHIFT or 2nd, then sin, cos or tan to get sin⁻¹, cos⁻¹ or tan⁻¹, and check the angle mode first. sin⁻¹(0.5) gives 30 in degree mode and about 0.5236 in radian mode. For arccsc, arcsec and arccot, take the reciprocal first: arcsec(2) = cos⁻¹(1/2) = 60°.

Can the iPhone calculator do inverse trig functions?

Yes. Turn the phone sideways (or open the scientific view), tap 2nd, and the sin, cos and tan keys change to sin⁻¹, cos⁻¹ and tan⁻¹. The Rad/Deg key sets the unit: tan⁻¹(1) shows 45 in degrees or about 0.7854 in radians.

Can you find inverse trig functions without a calculator?

Yes, for the special values in the table above. Recognise the ratio from a 30°-60°-90° or 45°-45°-90° triangle and pick the angle in the principal range: arcsin(1/2) = 30°, arccos(1/2) = 60°, arctan(1) = 45° and arccos(−1/2) = 120°.

How do you use inverse trig to find an angle in a triangle?

Choose the ratio that uses the two sides you know (SOH-CAH-TOA), then apply its inverse. With opposite side 5 and hypotenuse 10, θ = arcsin(5/10) = arcsin(0.5) = 30°. With opposite and adjacent both 7, θ = arctan(1) = 45°. The Right triangle tab does this and also finds the third side.

How do you get inverse trig answers in degrees?

Set the calculator to degree mode, or multiply a radian answer by 180/π. For example, arctan(2) ≈ 1.1071 rad, and 1.1071 × 180/π ≈ 63.43°.

What are the derivatives of inverse trig functions?

d/dx arcsin x = 1/√(1 − x²), d/dx arccos x = −1/√(1 − x²) and d/dx arctan x = 1/(1 + x²). At x = 1, the slope of arctan x is 1/(1 + 1) = 1/2.

Related calculators

• Unit circle calculator
• Trigonometry calculator
• Reference angle calculator
• Degrees to radians calculator
• Law of sines and cosines calculator