Arccos Calculator

Q: Is arccos the same as 1 divided by cosine?

No. They look alike but mean different things. arccos(x) or cos -1 (x) is the inverse function. It gives you an angle . 1/cos(x) is the reciprocal, called secant . It gives you a number . Example: arccos(0.5) = 60°, but 1/cos(0.5°) is about 1.00004. The small −1 in cos -1 is not an exponent.

Q: Why does a calculator show 1.047 instead of 60 for arccos(0.5)?

Your calculator is in radian mode . Both answers are correct, just in different units. 1.047198 rad = 60°. To switch, multiply radians by 180/π, or change the mode setting to DEG. If you need radians instead but keep seeing 60, you are in degree mode.

Q: Can arccos give a negative answer?

No. Arccos always returns an angle from 0° to 180° (0 to π radians). Even negative inputs give positive angles. For example, arccos(−0.5) = 120°, not −60°. Only arcsin and arctan can give negative angles.

Q: What is arccos of a negative number equal to?

Use this rule: arccos(−x) = 180° − arccos(x) In radians: arccos(−x) = π − arccos(x). Example: arccos(0.5) = 60°, so arccos(−0.5) = 180° − 60° = 120°. Negative inputs always land in the second quadrant, above 90°.

Q: Does arccos(cos(x)) always equal x?

Not always. cos(arccos(x)) = x for any x from −1 to 1. This one is always true. arccos(cos(x)) = x only when x is between 0° and 180°. Example: arccos(cos(200°)) = 160°, not 200°, because the answer must stay in the 0° to 180° range.

Q: What is the derivative of arccos(x)?

The derivative is: d/dx arccos(x) = −1 / √(1 − x²) It is the negative of the arcsin derivative. The minus sign shows arccos is always decreasing: as x grows, the angle shrinks. It works for x between −1 and 1 only.

Arccos Calculator

Introduction

This arccos calculator finds the angle when you know the cosine value. Arccos is also called inverse cosine, and it is written as arccos(x) or cos^-1(x). Type a number from −1 to 1, and you get the angle back.

The answer comes in three units: degrees, radians, and gradians. When the answer is a neat multiple of π, the calculator shows the exact form too, like π/3. You can also enter fractions such as 1/2 or -3/4.

Below the answer, you will see a step-by-step solution, a graph of y = arccos(x), and a table of common arccos values. Use them to check your work in trigonometry, geometry, or physics class.

How to use our Arccos Calculator

Type a cosine value between −1 and 1. The arccos calculator shows the angle in degrees, radians, and gradians, plus step-by-step math, a graph, and a table of common values.

Cosine value x: Type a number from −1 to 1 in the box. You can use a decimal like 0.5 or a fraction like 1/2 or -3/4. The answer updates as you type.

Calculate button: Click it to work out arccos(x) and show the steps. You can also just press Enter.

Reset button: Click it to clear your value and start again with the sample value 0.5.

Radian precision: Pick 4, 6, or 10 decimals to set how exact the radian answer is shown.

Y-axis unit: Pick degrees, radians, or gradians to change the unit on the arccos graph.

What Is Arccos (Inverse Cosine)?

Arccos is the inverse of cosine. Cosine takes an angle and gives you a number. Arccos does the opposite: you give it a number, and it gives you back the angle. It is also written as cos^-1(x).

Example: cos(60°) = 0.5, so arccos(0.5) = 60°.

The Rule for Inputs

You can only put in numbers from −1 to 1. That is the domain of arccos. Cosine never gives an answer outside that range, so arccos has no answer for numbers like 2 or −5.

The Range of Answers

Arccos always gives an angle between 0° and 180° (0 to π radians). This is called the principal value. A cosine value can match many angles, so math picks just one answer to keep things clear.

Where Arccos Comes From

In a right triangle, cosine is the adjacent side divided by the hypotenuse. If you know those two sides, arccos finds the missing angle:

angle = arccos(adjacent ÷ hypotenuse)

Degrees, Radians, and Gradians

Degrees (°): a full circle is 360°. Most common in school.
Radians (rad): a full circle is 2π. Used in higher math and physics.
Gradians (grad): a full circle is 400. Used in surveying.

To change degrees to radians, multiply by π/180. To change degrees to gradians, multiply by 10/9.

Values Worth Remembering

arccos(1) = 0° = 0 rad
arccos(√3/2) = 30° = π/6 rad
arccos(√2/2) = 45° = π/4 rad
arccos(0.5) = 60° = π/3 rad
arccos(0) = 90° = π/2 rad
arccos(−0.5) = 120° = 2π/3 rad
arccos(−1) = 180° = π rad

Where People Use It

Arccos helps find angles in triangles, ramps, and roof slopes. It is used in engineering, navigation, computer graphics, and robotics. It also shows up in the Law of Cosines, where you know all three sides of a triangle and need an angle.

Quick Tips

As x gets bigger, the angle gets smaller. Arccos is a decreasing function.
Negative inputs always give angles over 90°.
arccos(x) + arcsin(x) = 90° for any valid x.
Fractions like 1/2 or −3/4 work fine as inputs.

Formulas used

Inverse cosine (principal value in radians)

\theta = \arccos(x), \quad -1 \le x \le 1, \quad 0 \le \theta \le \pi

Radians to degrees

\theta_{\deg} = \theta_{\mathrm{rad}} \times \frac{180}{\pi}

Radians to gradians

\theta_{\mathrm{grad}} = \theta_{\mathrm{rad}} \times \frac{200}{\pi}

Degrees to radians

\theta_{\mathrm{rad}} = \theta_{\deg} \times \frac{\pi}{180}

Degrees to gradians

\theta_{\mathrm{grad}} = \theta_{\deg} \times \frac{10}{9}

Fraction input evaluated as decimal

x = \frac{a}{b}, \quad b \neq 0

Exact radian result as a multiple of pi

\theta_{\mathrm{rad}} = \frac{n\pi}{d}, \quad d = 1,2,\dots,12

Frequently asked questions

Is arccos the same as 1 divided by cosine?

No. They look alike but mean different things.

arccos(x) or cos^-1(x) is the inverse function. It gives you an angle.
1/cos(x) is the reciprocal, called secant. It gives you a number.

Example: arccos(0.5) = 60°, but 1/cos(0.5°) is about 1.00004. The small −1 in cos^-1 is not an exponent.

Why does a calculator show 1.047 instead of 60 for arccos(0.5)?

Your calculator is in radian mode. Both answers are correct, just in different units.

1.047198 rad = 60°. To switch, multiply radians by 180/π, or change the mode setting to DEG.

If you need radians instead but keep seeing 60, you are in degree mode.

How do you find every angle that has the same cosine value?

Arccos gives only one angle, from 0° to 180°. Cosine repeats, so there are many more.

If θ = arccos(x), then all answers are:

θ = ±arccos(x) + 360°n (or ±arccos(x) + 2πn in radians), where n is any whole number.

Example: cos θ = 0.5 gives 60°, 300°, 420°, −60°, and so on.

Can arccos give a negative answer?

No. Arccos always returns an angle from 0° to 180° (0 to π radians).

Even negative inputs give positive angles. For example, arccos(−0.5) = 120°, not −60°. Only arcsin and arctan can give negative angles.

What is arccos of a negative number equal to?

Use this rule:

arccos(−x) = 180° − arccos(x)

In radians: arccos(−x) = π − arccos(x).

Example: arccos(0.5) = 60°, so arccos(−0.5) = 180° − 60° = 120°. Negative inputs always land in the second quadrant, above 90°.

Does arccos(cos(x)) always equal x?

Not always.

cos(arccos(x)) = x for any x from −1 to 1. This one is always true.
arccos(cos(x)) = x only when x is between 0° and 180°.

Example: arccos(cos(200°)) = 160°, not 200°, because the answer must stay in the 0° to 180° range.

What is the derivative of arccos(x)?

The derivative is:

d/dx arccos(x) = −1 / √(1 − x²)

It is the negative of the arcsin derivative. The minus sign shows arccos is always decreasing: as x grows, the angle shrinks. It works for x between −1 and 1 only.

How do you find a triangle angle when you know all three sides?

Use the Law of Cosines, then arccos. For sides a, b, c with angle C across from side c:

C = arccos((a² + b² − c²) ÷ (2ab))

Example: a = 3, b = 4, c = 5 gives arccos((9 + 16 − 25) ÷ 24) = arccos(0) = 90°.

How do you find the angle between two vectors?

Divide the dot product by the two lengths, then take arccos:

θ = arccos((a · b) ÷ (|a| × |b|))

The result is always between 0° and 180°, which is exactly what you want for an angle between vectors. If the answer is 90°, the vectors are perpendicular.

What does the ACOS function do in Excel?

=ACOS(number) gives the inverse cosine, but the answer comes back in radians.

To get degrees, wrap it: =DEGREES(ACOS(0.5)) returns 60.

If the number is outside −1 to 1, Excel shows a #NUM! error.

How can you work out arccos without a calculator?

Learn the special triangle values. These cover most homework problems:

arccos(1) = 0°
arccos(√3/2) ≈ arccos(0.866) = 30°
arccos(√2/2) ≈ arccos(0.707) = 45°
arccos(1/2) = 60°
arccos(0) = 90°

For negatives, take 180° minus the matching positive answer.

What is the difference between arccos and arcsin?

Both take a number from −1 to 1 and give an angle, but their answer ranges differ.

arccos gives 0° to 180°. Use it with adjacent ÷ hypotenuse.
arcsin gives −90° to 90°. Use it with opposite ÷ hypotenuse.

They are linked: arccos(x) + arcsin(x) = 90° for every valid x.

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