Inverse Cosine

The inverse trigonometric function inverse cosine is very useful. It is the inverse function of the trigonometric function cosine and is expressed as cos-1(x) in mathematics (x). It’s worth noting that inverse cosine isn’t the same as cos x’s reciprocal. sin-1x, cos-1x, tan-1x, csc-1x, sec-1x, cot-1x are the six inverse trigonometric functions.

Using the value of the trigonometric ratio cos x, inverse cosine is used to get the angle measure. We will learn the formulae for the inverse cosine function.

What is Inverse Cosine?

The inverse function of the cosine function is inverse cosine. One of the most essential inverse trigonometric functions is this one. arccos x is another way to write cos inverse x.If y = cos x ⇒ x = cos-1(y )To understand how the inverse cosine function works, explore a few instances

cos 0 = 1 ⇒ 0 = cos-1 (1)

cos π/3 = 1/2 ⇒ π/3 = cos-1 (1/2)

cos π/2 = 0 ⇒ π/2 = cos-1 (0)

cos π = -1 ⇒ π = cos-1 (-1)

The cosine of an angle () in a right-angled triangle is the ratio of its adjacent side to the hypotenuse,that is, cos θ = (adjacent side) / (hypotenuse). By employing the definition of inverse cosine, θ = cos-1[ (adjacent side) / (hypotenuse) ].

In a right-angled triangle, the inverse cosine is employed to find the unknown angles.

Inverse Cosine’s Domain and range :

We know that the cosine function’s domain is R or all real numbers and that its range is [-1, 1]. If and only if a function f(x) is bijective, it has an inverse (one-one and onto). The inverse cosine cannot have R as its range since cos x is not a bijective function because it is not one-one. As a result, we must restrict the domain of the cosine function to make it one-one. An equivalent branch of inverse cosine may be obtained by restricting the domain of the cosine function to [0, π], [π, 2π], [-π, 0], and so on.

The domain of the cosine function is normally constrained to [0, π]and the range is [-1, 1]. As a result, the principal branch of cos inverse x with the range [0, π]is known. Because a function’s domain and range become the range and domain of its inverse function, the inverse cosine’s domain is [-1, 1] and its range is [0, π], hence cos inverse x is a function from [-1, 1] → [0, π].

Inverse Cos x Derivative

We’ll now use basic trigonometric formulae and identities to find the derivative of the inverse cosine function. Assume that y = cos-1x and that cos y = x. Using the chain rule, differentiate both sides of the equation cos y = x with respect to x.

cos y = x

⇒ d(cos y)/dx = dx/dx

⇒ -sin y dy/dx = 1

⇒ dy/dx = -1/sin y —- (1)

Since cos2y + sin2y = 1, we have sin y = √(1 – cos2y) = √(1 – x2) [Because cos y = x]

Putting sin y = √(1 – x2) in (1), we get :

dy/dx = -1/√(1 – x2)

Since x = -1, 1 equals 0 in the denominator √(1 – x2)the derivative is not defined, hence x cannot be both -1 and 1.

As a result derivative of cos inverse x is -1/√(1 – x2), where -1 < x < 1.

Inverse Cosine Integration

We will find the integral of inverse cosine, that is, ∫cos-1x dx using the integration by parts.

∫cos-1x = ∫cos-1x · 1 dx

Using integration by parts,

∫f(x) . g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) ∫g(x) dx) dx + C

Here f(x) = cos-1x and g(x) = 1.

∫cos-1x · 1 dx = cos-1x ∫1 dx – ∫ [d(cos-1x)/dx ∫1 dx]dx + C

∫cos-1x dx = cos-1x . (x) – ∫ [-1/√(1 – x²)] x dx + C

We will evaluate this integral ∫ [-1/√(1 – x²)] x dx using substitution method. Assume 1-x2 = u. Then -2x dx = du (or) x dx = -1/2 du.

∫cos-1x dx = x cos-1x – ∫(-1/√u) (-1/2) du + C

= x cos-1x – 1/2 ∫u-1/2 du + C

= x cos-1x – (1/2) (u1/2/(1/2)) + C

= x cos-1x – √u + C

= x cos-1x – √(1 – x²) + C

Therefore, ∫cos-1x dx = x cos-1x – √(1 – x²) + C

Inverse Cosine’s Characteristics

The inverse cosine function’s features and formulae are listed below. These are quite useful in solving trigonometry issues using cos inverse x.

1.cos(cos-1x) = x only when x ∈ [-1, 1](When x ∉ [-1, 1], cos(cos-1x) is NOT defined)

2.cos-1(cos x) = x, only when x ∈ [0, π](When x ∉ [0, π], apply the trigonometric identities to find the equivalent angle of x that lies in [0, π])

3.cos-1(-x) = π – cos-1x

4.cos-1(1/x) = sec-1x, when |x| ≥ 1

5.Sin-1x + cos-1x = π/2, when x ∈ [-1, 1]

6.d(cos-1x)/dx = -1/√(1 – x2), -1 < x < 1

7.∫cos-1x dx = x cos-1x – √(1 – x²) + C

Conclusion :

The inverse trigonometric function inverse cosine is very useful. It is the inverse function of the trigonometric function cosine and is expressed as cos-1(x) in mathematics (x).

In a right-angled triangle, the inverse cosine is employed to find the unknown angles.