Arccos Calculator
Use our free arccos calculator to instantly find the inverse cosine of any value between -1 and 1. Whether you need the result in degrees, radians, or in terms of pi, this calculator with arccos functionality delivers accurate results with a single click. No manual lookup required.
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What is Arccos?
Arccosine (written as arccos or cos⁻¹) is the inverse trigonometric function of cosine. In simple terms, while cosine tells you the ratio given an angle, arccos does the opposite. It finds the angle when the cosine ratio is known.
If cos(θ) = x, then arccos(x) = θ.
This function is also written as inv cos or acos, and you may see it on scientific calculators as the “cos⁻¹” button. It is important to note that cos⁻¹(x) in this context does not mean 1/cos(x). It specifically refers to the inverse cosine function. So, is arccos the same as cos⁻¹? Yes, they are the same thing, just different notations.
Domain and Range of Arccos
Understanding the domain of arccos and its range is essential before using the function:
- Domain of arccos: −1 ≤ x ≤ 1. The input value must lie between -1 and 1 (inclusive). Any value outside this range has no real arccosine.
- Range of arccos: 0° to 180° (or 0 to π in radians). These are the only output angles the arccos function produces.
So when someone asks what is the domain of arccos, the answer is [−1, 1], and the range of arccos is [0°, 180°].
How to Use This Arccos Calculator?
Using this arccos calc is straightforward:
- Enter a value between −1 and 1 in the input field.
- Click the “Calculate” button.
- The tool will instantly show the angle in degrees, radians, and in terms of pi.
- Click “Clear” to reset and start a new calculation.
This online tool works as a full arccosine calculator, an arc cosine calculator, and an opposite cosine calculator, all in one place.
How to Calculate Arccos Manually?
If you want to know how to find arccos without a calculator, follow these steps:
- Confirm your input value is within the valid domain: between −1 and 1.
- Apply the formula: θ = cos⁻¹(x)
- Look up the angle whose cosine equals your input value, either from memory, a trig table, or by checking the common values table below.
- Express the result in degrees (0°–180°) or radians (0–π).
For most practical problems, however, it is far easier to simply calculate arccos using this tool. Manual tables only cover common values, whereas this calculator handles any decimal input instantly.
Worked Example: arccos(0.5)
Problem: Find arccos(1/2)
Solution:
We need θ such that cos(θ) = 0.5
From trigonometry, we know: cos(60°) = 0.5, which is also cos(π/3) = 0.5
Therefore: arccos(0.5) = 60° = π/3 radians
Similarly, arccos(1/2) in terms of pi = π/3.
Common Arccos Values Table
The table below covers frequently searched values, including arccos 0, arccos 1/2, arccos 3/5, arccos 1/3, arccos 3/4, and more:
| x | arccos(x) in Degrees | arccos(x) in Radians | In Terms of Pi |
|---|---|---|---|
| −1 | 180° | π | π |
| −√3/2 ≈ −0.8660 | 150° | 5π/6 | 5π/6 |
| −√2/2 ≈ −0.7071 | 135° | 3π/4 | 3π/4 |
| −0.5 | 120° | 2π/3 | 2π/3 |
| 0 | 90° | π/2 | π/2 |
| 1/3 ≈ 0.3333 | ≈ 70.53° | ≈ 1.2310 rad | — |
| 0.5 (= 1/2) | 60° | π/3 | π/3 |
| 3/5 = 0.6 | ≈ 53.13° | ≈ 0.9273 rad | — |
| 2/3 ≈ 0.6667 | ≈ 48.19° | ≈ 0.8411 rad | — |
| √2/2 ≈ 0.7071 | 45° | π/4 | π/4 |
| 3/4 = 0.75 | ≈ 41.41° | ≈ 0.7227 rad | — |
| 4/5 = 0.8 | ≈ 36.87° | ≈ 0.6435 rad | — |
| √3/2 ≈ 0.8660 | 30° | π/6 | π/6 |
| 1 | 0° | 0 | 0 |
For specific cases like arccos 1/2 in degrees, the answer is 60°. For arccos 0 in degrees, the answer is 90°. For arccos 1 3 in degrees, use the calculator above for the precise decimal result.
Arccos Identities and Properties
These identities are useful in trigonometry and calculus:
- cos(arccos(x)) = x : Applying cosine to its inverse returns the original value
- arccos(−x) = π − arccos(x) : Reflection property across 90°
- sin(arccos(x)) = √(1 − x²) : Relates arccosine to sine
- arccos(x) + arcsin(x) = π/2 : Complementary relationship
- arccos(α) + arccos(β) = arccos(αβ − √((1−α²)(1−β²))) : Addition formula
One common question is: is arccos the same as cos 1? The answer is no, cos(1) is simply the cosine of 1 radian. Arccos, or cos⁻¹, is a function that reverses cosine. The notation cos⁻¹(x) can be misleading, which is why many prefer to write it as arccos(x) or acos(x).
Similarly, is arccos same as cos⁻¹? Yes, they are identical, just two different ways of writing the same inverse trigonometric function.
How to Do Arccos on a Calculator (Scientific/Physical)
If you are wondering how to do arccos on a calculator (a physical scientific calculator), here is how:
- Press the “2nd” or “Shift” key (depending on your calculator model).
- Then press the “cos” key. This activates the cos⁻¹ or arccos function.
- Enter your value and press “=” or “Enter”.
Make sure your calculator is set to the correct mode, degrees or radians, before performing the calculation. This applies whether you are using a Casio, Texas Instruments (TI), or any other scientific calculator.
For a faster, browser-based experience, this online arccos in calculator tool requires no mode-switching and shows results in all three formats at once.
Arccos Graph
The graph of y = arccos(x) is a decreasing curve starting at the point (−1, 180°) and ending at (1, 0°). Key visual characteristics:
- The curve passes through (0, 90°), confirming arccos 0 = 90°
- It is strictly decreasing from left to right
- The graph is bounded between y = 0° and y = 180°
This shape makes it easy to see why the domain and range of arccos are restricted: the function only produces one unique angle per input value, keeping it well-defined and invertible.
Why Use This Arccos Calculator?
- Covers both degrees and radians output simultaneously
- Handles all values within the valid domain for arccos, including negative inputs
- Works as a calculator with arccos, no need to switch between tools
- Displays results in terms of pi for exact fractional answers
- Faster and more reliable than using arccos on a physical calculator for one-off calculations
- Free, instant, and works directly in your browser
Whether you are a student solving a trigonometry problem, an engineer working with angles, or just need to quickly calculate arc cosine of a specific ratio, this tool has you covered.
Frequently Asked Questions
What is arccos0?
arccos(0) = 90° or π/2 radians.
What is arccos of 0?
arccos of 0 is 90 degrees.
What is arccos 1/2 in degrees?
arccos(1/2) = 60°
What is arccos 1/2 in terms of pi?
arccos(1/2) = π/3
What is arccos 3/5 in degrees?
arccos(3/5) ≈ 53.13°
What is arccos 2/3 in degrees?
arccos(2/3) ≈ 48.19°
What is arccos 1/3 in degrees?
arccos(1/3) ≈ 70.53°
What is arc cos 4/5?
arccos(4/5) = arccos(0.8) ≈ 36.87°
What is arc cos 0?
arc cos(0) = 90°
What is arccos of cos(x)?
arccos(cos(x)) = x, as long as x is within the range [0°, 180°].
What is arccos1?
arccos(1) = 0° or 0 radians
What is arcos1?
0 degrees
Can I use this as an inv cos calculator?
Yes. This tool functions as a full inverse cosine calculator or acos calculator. Enter any value from −1 to 1 and get the angle instantly.
Can I calculate arc cosine of 1?
Yes. The arc cosine of 1 is 0°.