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Arccos Calculator

Provide required entities (decimal or fraction), and the arccos calculator will instantly compute inverse cosine values for them in radians or degrees.

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An online arccos calculator allows you to calculate \( \arccos(x) \), the inverse of the cosine function. It supports decimal inputs (e.g., 0.5, -0.5) and displays results in both degrees and radians. This tool is helpful for students, teachers, and anyone working with trigonometry.

What is Arccos?

The arccosine of \(x\), written as \( \arccos(x) \) or \( \cos^{-1}(x) \), is the inverse cosine function. It returns the angle \(y\) such that:

\( \cos y = x \), where \( -1 \le x \le 1 \).

In other words:

\( \arccos(x) = y \quad \text{if and only if} \quad \cos(y) = x \)

Arccos Formula

The inverse cosine equation is straightforward:

\( \arccos(x) = \cos^{-1}(x) \)

How to Calculate Arccos

Example: Find the arccos of 0.5.

Given \( \cos 60^\circ = 0.5 \)
Apply the inverse cosine formula: \( \arccos(x) = \cos^{-1}(x) \)
Result: \( \arccos(0.5) = 60^\circ \) or \( \pi/3 \) radians.

Using Arccos to Find an Angle in a Triangle

For a right triangle, the formula is:

\( \Theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) \)

Example:

Let \( a = 30 \) (adjacent), \( c = 26 \) (hypotenuse). Then:

\( \cos(\theta) = \frac{a}{c} = \frac{30}{26} = 0.866 \)

Using the inverse function:

\( \arccos(0.866) = 30^\circ \approx 0.5236 \, \text{radians} \)

For more trigonometric calculations, see the online arcsin calculator.

Common Arccos Values Table

x	arccos(x) (°)	arccos(x) (rad)
-1	180°	π
-√3/2	150°	5π/6
-√2/2	135°	3π/4
-1/2	120°	2π/3
0	90°	π/2
1/2	60°	π/3
√2/2	45°	π/4
√3/2	30°	π/6
1	0°	0

Graph of Arccos

The graph visualizes the one-to-one inverse of the cosine function, showing how arccos maps values from [-1, 1] to [0, π]:

Arccos Graph

How the Arccos Calculator Works

Input:

Enter a real number \( x \) in the domain \( -1 \le x \le 1 \).
Select the desired number of decimal places.
Click "Calculate".

Output:

Angle in degrees and radians.
Optional: gradian, turns, arc minutes/seconds, milliradians, microradians, and π-radians.
Detailed calculation summary.

Also see the unit circle calculator for sine, cosine, and tangent values.

FAQs

Do Arccos and Cos cancel each other?

Yes, but only for \( x \in [-1, 1] \). Cos(arccos(x)) = x.

Is SEC the inverse of Cos?

No. Secant is the reciprocal of cosine: hypotenuse / adjacent.

What is the inverse of Tan?

The inverse of tangent is arctangent (arctan).

Is Arcsin the inverse of Sin?

Yes, arcsin returns the angle whose sine equals the given number.

References

Wikipedia: Inverse Trigonometric Functions
The Math Page: Inverse Cosine Function
CliffsNotes: Principal values and inverse cosine

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