Inverse Cosine: Overview, Questions, Preparation

The inverse of any function f is defined as f-1. Similarly, cosine, which is one of the essential trigonometric functions obtained by dividing the base by hypotenuse, also has an inverse. The inverse cosine is used to get trigonometry concepts data about the angle directly.

These are also called Arc Functions because they give the length of the arc for a given value of trigonometric functions. So cos–1 can also be written as arccos(x)

For an angle, α = cos-1 (base / hypotenuse)

The domain of cos is R[-1, 1]

cos–1 is a function whose domain is [–1, 1] and range could be any of the intervals [–π, 0], [0, π], [π, 2π] etc. 

The branch with range [0, π] is called the principal value branch of the function cos–1. 

Hence we denote the domain of cos–1 as [–1, 1] → [0, π].

Another point to remember is if cos–1 x = y, then cos y = x. The cos–1 and cos negates, so cos–1 (cos x) = x

While solving sums, make a note of the following:

The domain and range of arccosine function is denoted as;

For cos–1 x = y,

Domain: −1 ≤ x ≤ 1

Range: 0 ≤ y ≤ π

Example: Find the principal value of cos–1 1/2.

We know that the range is between 0 to π, and we know cos π/3 = ½, 

hence cos–1 1/2 is π/3.

1. cos–1 (1/x) = sec –1x, x ≥ 1 or x ≤ – 1

As secant is technically the reciprocal of cosine, this is carried onto their inverses too.

2. cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1]

To prove, let us take cos–1 (–x) = y then

 –x = cos y

x = – cos y = cos (π – y) 

Therefore cos–1 x = π – y = π – cos–1 (–x) 

Hence cos–1 (–x) = π – cos–1 x

3. sin–1 x + cos–1 x = π/2 , x ∈ [– 1, 1]

Let sin–1 x = y. Then x = sin y = cos (π/2 - y)

cos–1 x = (π/2 - y) = π/2 - sin–1 x

Hence sin–1 x + cos–1 x = π/2.

Students are introduced to this topic in Class 12 to understand the concepts of trigonometry. By learning the inverse functions, they learn to solve both sides of the equations, allowing them to comprehend complex derivations and answer in-depth questions. This helps them in their national exams and above all, will enable them to simplify massive equations to simpler forms to solve problems. This is applied in Physics for proofs and solving engineering-related questions.

1. Solve cos (cos-1 7π/2) 

We know that cos (cos -1) x = x, hence cos (cos-1 7π/2) = 7π/2

2. Find cos–1 (-1/2)   

 cos–1 (–x) = π – cos–1 x, hence cos–1 (–1/2) = π – cos–1 1/2

= π – π/3

= 2π/3

3. Solve cos-13/2

cos-13/2 = π/6

4. Let the value of the base is √2, and the hypotenuse is 2. Find the value of angle α?

By the inverse cosine formula we know,

α = cos-1 (base / hypotenuse)

α = cos-1(√2 /2)

Therefore, α = π/4.

Q" What is the principal branch value of a trigonometric function?

Q: Is cos-1 x = (cos x)-1?

Q: What is the domain and range for cos-1 x?

Q: How can you write cos-1 x in terms of sin-1 x?

Q: What is cos–1 (cos x)?