Inverse cosine function and its properties

Suppose the cosine of an angle is y i.e. cos x = y, then its inverse cosine function can be written as cos-1y = x. It is also symbolized as arccos(y). So if we have the cosine value and need to find the angle, then inverse cosine function is the solution. It can be noted that the output values of all inverse trig functions are angles, it may be in degrees, grades or radians. So lets have a look at the graph of inverse cosine function and list out important properties like the domain and the range.

Graph of inverse cosine function with important properties

The graph of inverse cosine function is shown below. It looks like a reflected image of the cosine function graph.
graph of inverse cosine function

Some important properties are listed below which will make the topic more clear.

  • The domain of inverse cosine function is between (-1, 1).

  • The range of the inverse function is (0, π).

  • cos-1(-y) = π – cos-1(y).

  • cos-1(y) = sec-1(1/y).

Since cosine of an angle is the ratio of length of adjacent to that of hypotenuse, we can say that the inverse cosine function cos-1(adjacent/hypotenuse) gives the required cosine angle. The derivative of inverse cos function of x is given by -(1/√(1-x2)) while its integration is given by xsin-1x√(1-x2) + c where c is the constant. This information is enough to let us move to the examples.

(1) Find the angle θ for a right triangle with length of adjacent side 3 cm and that of opposite side 4 cm:

By pythagoras theorem, we have length of hypotenuse h = √(32 + 42) = 5. Now cosine of an angle is cosθ = adjacent / hypotenuse = 3 / 5. Taking the inverse on both sides, θ = cos-1(3/5) = 53.13 degrees.

(2) Find cos-1(-1/2):

We know that cos-1(-y) = π – cos-1(y). Hence cos-1(-1/2) = π – cos-1(1/2) = π – π/3 = 2π/3 rad.

The inverse cosine functions can be directly applied for solving integration as well as differentiation problems. They also find wide applications in space, research and engineering categories.

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