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Inverse Cosine

Inverse Cosine
cos-1
Cos-1
arccos
Arccos

The inverse function of cosine.

Basic idea: To find cos-1 (½), we ask "what angle has cosine equal to ½?" The answer is 60°. As a result we say cos-1 (½) = 60°. In radians this is cos-1 (½) = π/3.

More: There are actually many angles that have cosine equal to ½. We are really asking "what is the simplest, most basic angle that has cosine equal to ½?" As before, the answer is 60°. Thus cos-1 (½) = 60° or cos-1 (½) = π/3.

Details: What is cos-1 (–½)? Do we choose 120°, –120°, 240°, or some other angle? The answer is 120°. With inverse cosine, we select the angle on the top half of the unit circle. Thus cos-1 (–½) = 120° or cos-1 (–½) = 2π/3.

In other words, the range of cos-1 is restricted to [0, 180°] or [0, π].

Note: arccos refers to "arc cosine", or the radian measure of the arc on a circle corresponding to a given value of cosine.

Technical note: Since none of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent are one-to-one, their inverses are not functions. Each trig function can have its domain restricted, however, in order to make its inverse a function. Some mathematicians write these restricted trig functions and their inverses with an initial capital letter (e.g. Cos or Cos-1). However, most mathematicians do not follow this practice. This website does not distinguish between capitalized and uncapitalized trig functions.

 

Unit circle showing range of cos⁻¹: from 0 (right) to π (left), spanning the upper semicircle, with π/2 at top.

 

See also

Inverse trigonometry, inverse trig functions, interval notation

Key Formula

θ=cos⁡−1(x)where −1≤x≤1 and 0≤θ≤π\theta = \cos^{-1}(x) \quad \text{where } -1 \le x \le 1 \text{ and } 0 \le \theta \le \pi
Where:
  • θ\theta = The output angle, measured in radians (or degrees from 0° to 180°)
  • xx = The input value (the known cosine), which must lie between −1 and 1 inclusive

Worked Example

Problem: Find the exact value of cos⁻¹(√3/2).
Step 1: Ask: which angle θ in [0, π] satisfies cos θ = √3/2?
cos⁡θ=32\cos\theta = \frac{\sqrt{3}}{2}
Step 2: Recall the unit circle values. Cosine equals √3/2 at 30° (π/6).
cos⁡ ⁣(π6)=32\cos\!\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
Step 3: Verify that π/6 lies in the required range [0, π]. It does, so this is the answer.
0≤π6≤π✓0 \le \frac{\pi}{6} \le \pi \quad \checkmark
Answer: cos⁻¹(√3/2) = π/6, or equivalently 30°.

Another Example

This example uses a negative input, showing how inverse cosine returns a second-quadrant angle rather than a negative angle or a reflex angle.

Problem: Find the exact value of cos⁻¹(−√2/2).
Step 1: Ask: which angle θ in [0, π] satisfies cos θ = −√2/2?
cos⁡θ=−22\cos\theta = -\frac{\sqrt{2}}{2}
Step 2: Recall that cos(45°) = √2/2. Because the input is negative, the angle must be in the second quadrant (between 90° and 180°).
θ=180°−45°=135°\theta = 180° - 45° = 135°
Step 3: Convert to radians: 135° = 3π/4.
θ=3π4\theta = \frac{3\pi}{4}
Step 4: Confirm: cos(3π/4) = −√2/2, and 3π/4 is within [0, π].
cos⁡ ⁣(3π4)=−22✓\cos\!\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \quad \checkmark
Answer: cos⁻¹(−√2/2) = 3π/4, or equivalently 135°.

Frequently Asked Questions

What is the difference between arccos and cos⁻¹?
They are two notations for the same function. arccos(x) and cos⁻¹(x) both return the angle whose cosine is x. The notation cos⁻¹ does not mean 1/cos (which is secant); it specifically denotes the inverse function.
Why is the range of inverse cosine [0, π] and not [−π, π]?
Cosine is not one-to-one over all real numbers, so we must restrict its domain to make the inverse a proper function. The interval [0, π] is chosen because cosine takes every value from −1 to 1 exactly once on that interval. This restriction corresponds to the top half of the unit circle.
How do you find inverse cosine on a calculator?
Most scientific calculators have a cos⁻¹ or arccos button, often accessed by pressing a "2nd" or "shift" key before the cos button. Make sure you set your calculator to degree mode or radian mode depending on which unit you need. Enter the value (between −1 and 1) and press the arccos key.

Inverse Cosine (arccos) vs. Inverse Sine (arcsin)

Inverse Cosine (arccos)Inverse Sine (arcsin)
Notationcos⁻¹(x) or arccos(x)sin⁻¹(x) or arcsin(x)
Domain (input)[−1, 1][−1, 1]
Range (output)[0, π] or [0°, 180°][−π/2, π/2] or [−90°, 90°]
Output quadrantsQuadrant I or II (top half of unit circle)Quadrant I or IV (right half of unit circle)
Value at 0arccos(0) = π/2arcsin(0) = 0
Identity linkarccos(x) + arcsin(x) = π/2arcsin(x) + arccos(x) = π/2

Why It Matters

Inverse cosine appears constantly in trigonometry, physics, and engineering whenever you need to recover an angle from a known ratio. For example, if you know the adjacent side and hypotenuse of a right triangle, arccos gives you the angle directly. It is also central to the law of cosines, where solving for an angle in any triangle requires applying cos⁻¹ to both sides of the equation.

Common Mistakes

Mistake: Confusing cos⁻¹(x) with 1/cos(x).
Correction: The superscript −1 here means the inverse function, not a reciprocal. The reciprocal of cosine is secant: sec(x) = 1/cos(x). Meanwhile, cos⁻¹(x) returns an angle.
Mistake: Giving an angle outside the range [0, π].
Correction: Inverse cosine always returns a value between 0 and π (0° and 180°). For instance, cos⁻¹(−1/2) = 2π/3, not −2π/3 or 4π/3, even though cosine of those other angles also equals −1/2.

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