Cos Inverse Calculator

The cos inverse is the inverse of the cosine function (no surprises here). That is, the cos inverse finds the angle that produces a particular cosine value. We denote this function by arccos, and we have the following formula: arccos(x) = y if and only if x = cos(y) for x ∈ [-1, 1].

The domain of the cos inverse is [-1, 1]. In other words, you can calculate cos inverse only for values between -1 and 1. Why? Recall that the range of a function becomes the domain of its inverse. As the cosine has values between -1 and 1, this interval is the domain of the cos inverse.

The range of cos inverse is the interval [0, π] in radians, or [0,180°] in degrees. Why? As you most certainly remember, the cosine is a periodic function; in particular, it is many-to-one. We cannot invert such functions! First, we must restrict it to an interval where the function is one-to-one. For the cosine, we most often restrict to the interval [0, π]. This is why this interval is the range of the cos inverse.

To compute the cos inverse of a negative number -x: Determine the absolute value of your number (i.e., remove the minus sign): x. Compute the cos inverse of the value from Step 1: arccos(x). You may want to use an online cos inverse calculator to do that. Subtract the value obtained in Step 2 from π, i.e., compute π - arccos(x). This is your answer! We've used here the formula arccos(-x) = π - arccos(x).

The answer is 90°, that is, π/2 rad. One way to get this result is to reformulate the question and ask what is the angle that lies in the interval [0, π] and its cosine is zero? It may help to sketch the cosine graph (the usual wavy stuff) and check where it crosses the x-axis. A quick look at this graph will tell you that the angle we're looking for is indeed the correct angle.