ACOS Calculator

July 5, 2025

0 Views

ACOS (Arccosine) Calculator

This tool calculates the arccosine (inverse cosine) of a number, finding the angle whose cosine is the given input value.

Enter a single number between -1 and 1 (inclusive). The calculator will return the corresponding angle in both radians and degrees.

Enter a Value for ACOS

Input Value (between -1 and 1): Enter a decimal number between -1 and 1.

Understanding ACOS (Arccosine)

What is ACOS?

ACOS, or arccosine, is the inverse function of the cosine (COS) function. While COS takes an angle and returns the ratio of the adjacent side to the hypotenuse in a right-angled triangle, ACOS takes that ratio (a number between -1 and 1) and returns the corresponding angle.

Mathematically, if cos(θ) = x, then acos(x) = θ.

It's often written as acos(x), arccos(x), or cos⁻¹(x).

Input and Output Range

The domain (valid input values) for ACOS is limited to numbers from **-1 to 1**, inclusive. This is because the output of the cosine function can only ever be within this range.

The principal range (output values) for ACOS is typically defined as angles from **0 to π radians** (or **0 to 180 degrees**). This specific range ensures that for every valid input between -1 and 1, there is a unique arccosine value.

Units: Radians vs. Degrees

Angles can be measured in radians or degrees:

**Radians:** A unit based on the radius of a circle. 2π radians is a full circle. ACOS functions in programming languages (like JavaScript's `Math.acos()`) typically return values in radians. The output will be between 0 and π.
**Degrees:** A unit based on dividing a circle into 360 parts. ACOS in degrees will return a value between 0° and 180°.

This calculator provides results in both units for your convenience.

ACOS Calculation Examples

Here are some common ACOS calculations:

Example 1: ACOS(1)

Scenario: Find the angle whose cosine is 1.

1. Input Value: 1

2. Calculation: acos(1)

3. Result (Radians): 0 radians (since cos(0 radians) = 1)

4. Result (Degrees): 0 degrees (since cos(0 degrees) = 1)

Example 2: ACOS(0)

Scenario: Find the angle whose cosine is 0.

1. Input Value: 0

2. Calculation: acos(0)

3. Result (Radians): π/2 radians ≈ 1.5708 radians (since cos(π/2) = 0)

4. Result (Degrees): 90 degrees (since cos(90°) = 0)

Example 3: ACOS(-1)

Scenario: Find the angle whose cosine is -1.

1. Input Value: -1

2. Calculation: acos(-1)

3. Result (Radians): π radians ≈ 3.1416 radians (since cos(π) = -1)

4. Result (Degrees): 180 degrees (since cos(180°) = -1)

Example 4: ACOS(0.5)

Scenario: Find the angle whose cosine is 0.5 (or 1/2).

1. Input Value: 0.5

2. Calculation: acos(0.5)

3. Result (Radians): π/3 radians ≈ 1.0472 radians (since cos(π/3) = 0.5)

4. Result (Degrees): 60 degrees (since cos(60°) = 0.5)

Example 5: ACOS(-0.5)

Scenario: Find the angle whose cosine is -0.5 (or -1/2).

1. Input Value: -0.5

2. Calculation: acos(-0.5)

3. Result (Radians): 2π/3 radians ≈ 2.0944 radians (since cos(2π/3) = -0.5)

4. Result (Degrees): 120 degrees (since cos(120°) = -0.5)

Example 6: ACOS(√2 / 2) ≈ ACOS(0.7071)

Scenario: Find the angle whose cosine is square root of 2 divided by 2.

1. Input Value: √2 / 2 ≈ 0.7071

2. Calculation: acos(√2 / 2)

3. Result (Radians): π/4 radians ≈ 0.7854 radians (since cos(π/4) = √2 / 2)

4. Result (Degrees): 45 degrees (since cos(45°) = √2 / 2)

Example 7: ACOS(-√2 / 2) ≈ ACOS(-0.7071)

Scenario: Find the angle whose cosine is negative square root of 2 divided by 2.

1. Input Value: -√2 / 2 ≈ -0.7071

2. Calculation: acos(-√2 / 2)

3. Result (Radians): 3π/4 radians ≈ 2.3562 radians (since cos(3π/4) = -√2 / 2)

4. Result (Degrees): 135 degrees (since cos(135°) = -√2 / 2)

Example 8: ACOS(√3 / 2) ≈ ACOS(0.8660)

Scenario: Find the angle whose cosine is square root of 3 divided by 2.

1. Input Value: √3 / 2 ≈ 0.8660

2. Calculation: acos(√3 / 2)

3. Result (Radians): π/6 radians ≈ 0.5236 radians (since cos(π/6) = √3 / 2)

4. Result (Degrees): 30 degrees (since cos(30°) = √3 / 2)

Example 9: ACOS(-√3 / 2) ≈ ACOS(-0.8660)

Scenario: Find the angle whose cosine is negative square root of 3 divided by 2.

1. Input Value: -√3 / 2 ≈ -0.8660

2. Calculation: acos(-√3 / 2)

3. Result (Radians): 5π/6 radians ≈ 2.6180 radians (since cos(5π/6) = -√3 / 2)

4. Result (Degrees): 150 degrees (since cos(150°) = -√3 / 2)

Example 10: ACOS(0.2)

Scenario: Find the angle whose cosine is 0.2.

1. Input Value: 0.2

2. Calculation: acos(0.2)

3. Result (Radians): ≈ 1.3694 radians

4. Result (Degrees): ≈ 78.46 degrees

Frequently Asked Questions about ACOS

1. What is ACOS?

ACOS (Arccosine) is the inverse trigonometric function of cosine. It answers the question, "What angle has this cosine value?"

2. What values can I input into the ACOS calculator?

You can only input values between -1 and 1, inclusive. These are the only possible outputs of the standard cosine function, so they are the only valid inputs for its inverse.

3. What is the output range of ACOS?

Standard ACOS functions return an angle between 0 and π radians (or 0° and 180°) inclusive. This is known as the principal value of the arccosine.

4. Why does the calculator show results in both radians and degrees?

Radians and degrees are the two most common units for measuring angles. Depending on your context (mathematics, physics, engineering), you might need the result in one unit or the other. Standard programming functions usually return radians.

5. How is ACOS related to COS?

They are inverse functions. If you calculate the cosine of the angle returned by ACOS(x), you will get back the original value x (within the valid range). For example, cos(acos(0.5)) = 0.5.

6. What is ACOS(0)?

ACOS(0) is the angle whose cosine is 0. This is π/2 radians or 90 degrees.

7. What is ACOS(1)?

ACOS(1) is the angle whose cosine is 1. This is 0 radians or 0 degrees.

8. What is ACOS(-1)?

ACOS(-1) is the angle whose cosine is -1. This is π radians or 180 degrees.

9. Why did I get an error message?

An error occurs if your input is not a valid number or is outside the allowed range of -1 to 1. ACOS is undefined for numbers outside this domain.

10. What are some real-world uses of ACOS?

ACOS is used in various fields involving angles and ratios, such as trigonometry, physics (e.g., calculating angles in vectors or waves), engineering, computer graphics, and calculating distances on a sphere (like latitude/longitude). It helps find angles when you know the lengths of sides in a right-angled triangle or similar relationships.