Inverse Cosine

Inverse Cosine is one of the Trigonometric functions. Each trigonometric function has an inverse function of it, whether it sine, cosine, tangent, secant, cosecant and cotangent. These functions are also widely used, apart from the trigonometric formulas, to solve many problems in Maths. These functions are also called as Arc Functions because they give the length of the arc for a given value of trigonometric functions. They are usually written as arcsin(x), arccos(x), arctan(x), etc.

Inverse Cosine Definition

In a right-angled triangle, the cosine function is defined as the ratio of the length of base or adjacent side of the triangle(adjacent to angle) to that of the hypotenuse(the longest side) of the triangle. The Inverse Cosine function is the inverse of the Cosine function and is used to obtain the value of angles for a right-angled triangle.

Inverse Cosine Function Graph

The inverse of the cosine function is also called as “Arc Function” and is denoted as Arccos or Arccosine (acos). The graph of Arccosine function is given below;

Where y=cos^-1 x(arccosine of x)

Also, the domain and range of arccosine function is denoted as;

Domain: −1 ≤ x ≤ 1

Range: 0 ≤ y ≤ π

Similarly, we can define other arc functions like;

Arcsine functions(inverse of sine function)

y = sin^-1 x

Arctangent function(inverse of tangent function)

y = tan^-1 x

Arccotangent function(inverse of cotangent function)

y = cot^-1 x

Arcsecant function(inverse of secant function)

y = sec^-1 x

And Arccosecant function(inverse of cosecant function)

y = cosec^-1 x

Inverse Cosine Formula

As we know, the formula for cosine function, according to the above-given diagram is,

Cos α = Base/Hypotenuse

I.e. Cos α =b/h

Therefore, the inverse cosine formula becomes;

cos-1(Base/Hypotenuse) = α

Hence, we get the value for the angle α here.

Properties of Inverse Cosine

1. cos^-1x = sec^-1x, x ≥ 1 or x ≤ – 1
2. cos^-1 (–x) = π – cos^-1 x, x ∈ [– 1, 1]
3. sin^-1 x + cos^-1 x = π/2 , x ∈ [– 1, 1]

Inverse Cosine Examples

Problem: Let the value of the base is √3 and the hypotenuse is 2. Find the value of angle α?

Solution: By the inverse cos formula we know,

α = cos^-1(Base/Hypotenuse)

α = cos^-1(√3 /2)

Therefore, α = 30°

Problem: Find angle α, if the value of the base or adjacent side is 1 and the value of the hypotenuse is 2.

Solution: We know, cos α=Base/Hypotenuse=b/h

And cos^-1(Base/Hypotenuse) = α

cos^-1(1/2)= α

Therefore, α = 60°