Cos Inverse -1 = 180°
The exact value of cos⁻¹(-1), also written arccos(-1), is 180° or π radians.
What is cos inverse of -1?
cos⁻¹(-1) = 180° = π radians. The inverse cosine function asks: “Which angle has a cosine of -1?” On the principal range of arccosine, the answer is 180°.
You may also see the expression written as arccos(-1). In this context, cos⁻¹ means inverse cosine, not 1/cosine.
Why is arccos(-1) equal to 180°?
On the unit circle, the point at 180° (π radians) is directly opposite the starting point at 0°. Its x-coordinate is -1, and cosine is the x-coordinate of a point on the unit circle. Therefore:
cos(180°) = -1
Applying the inverse function gives:
arccos(-1) = 180°
Answer in degrees and radians
| Form | Answer |
|---|---|
| Degrees | 180° |
| Radians | π |
| Decimal radians | 3.14159265… |
The principal range of arccos
The principal value of arccos(x) is restricted to 0° through 180°, or 0 through π radians. This restriction makes inverse cosine a well-defined function even though cosine repeats its values around the unit circle.
| x | arccos(x) |
|---|---|
| 1 | 0° = 0 |
| √3/2 | 30° = π/6 |
| √2/2 | 45° = π/4 |
| 1/2 | 60° = π/3 |
| 0 | 90° = π/2 |
| -1/2 | 120° = 2π/3 |
| -√2/2 | 135° = 3π/4 |
| -√3/2 | 150° = 5π/6 |
| -1 | 180° = π |
Is 180° the only angle whose cosine is -1?
It is the only principal arccos answer. If you are solving the equation cos(θ) = -1 over all real angles, the complete family of solutions is θ = 180° + 360°k, where k is any integer. In radians, θ = π + 2πk.
How to calculate cos inverse values
For an inverse cosine problem, enter a value between -1 and 1 into an arccos calculator. For example, arccos(0) = 90°, arccos(1) = 0°, and arccos(-1) = 180°. For non-special values, a calculator gives the corresponding angle numerically.
Use the Trigonometry Calculator to calculate sin, cos, tan, arcsin, arccos, and arctan in degrees or radians.
Common mistakes
- Confusing inverse cosine with reciprocal: cos⁻¹(x) means arccos(x) when used as inverse-function notation; it does not mean 1/cos(x).
- Using an input outside [-1, 1]: real-valued arccos is defined only for inputs from -1 to 1.
- Mixing degrees and radians: 180° and π radians are the same angle, but they are different representations.
- Listing every periodic solution: arccos returns the principal value. A full equation such as cos(θ) = -1 has infinitely many periodic solutions.
Quick answers
What is cos inverse of -1?
180° or π radians.
What is arccos(-1) in radians?
π radians, approximately 3.14159265.
Why is cos 180° equal to -1?
At 180° on the unit circle, the point is (-1, 0), and cosine is the x-coordinate.
Can arccos return an angle greater than 180°?
Not as its principal value. The principal range is 0° to 180°. A separate equation-solving problem can have additional periodic solutions.