Inverse Cosine
Inverse cosine is an important inverse trigonometric function. Mathematically, it is written as cos^{1}(x) and is the inverse function of the trigonometric function cosine, cos(x). An important thing to note is that inverse cosine is not the reciprocal of cos x. There are 6 inverse trigonometric functions as sin^{1}x, cos^{1}x, tan^{1}x, csc^{1}x, sec^{1}x, cot^{1}x.
Inverse cosine is used to determine the measure of angle using the value of the trigonometric ratio cos x. In this article, we will understand the formulas of the inverse cosine function, its domain and range, and hence, its graph. We will also determine the derivative and integral of cos inverse x to understand its properties better.
What is Inverse Cosine?
Inverse cosine is the inverse function of the cosine function. It is one of the important inverse trigonometric functions. Cos inverse x can also be written as arccos x. If y = cos x ⇒ x = cos^{1}(y). Let us consider a few examples to see how the inverse cosine function works.
 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)
In a rightangled triangle, the cosine of an angle (θ) is the ratio of its adjacent side to the hypotenuse, that is, cos θ = (adjacent side) / (hypotenuse). Using the definition of inverse cosine, θ = cos^{1}[ (adjacent side) / (hypotenuse) ].
Thus, the inverse cosine is used to find the unknown angles in a rightangled triangle.
Domain and Range of Inverse Cosine
We know that the domain of the cosine function is R, that is, all real numbers and its range is [1, 1]. A function f(x) has an inverse if and only if it is bijective(oneone and onto). Since cos x is not a bijective function as it is not oneone, the inverse cosine cannot have R as its range. Hence, we need to make the cosine function oneone by restricting its domain. The domain of the cosine function can be restricted to [0, π], [π, 2π], [π, 0], etc. and get a corresponding branch of inverse cosine.
The domain of cosine function is restricted to [0, π] usually and its range remains as [1, 1]. Hence, the branch of cos inverse x with the range [0, π] is called the principal branch. Since the domain and range of a function become the range and domain of its inverse function, respectively, the domain of the inverse cosine is [1, 1] and its range is [0, π], that is, cos inverse x is a function from [1, 1] → [0, π].
Graph of Inverse Cosine
Since the domain and range of the inverse cosine function are [1, 1] and [0, π], respectively, we will plot the graph of cos inverse x within the principal branch. As we know the values of the cosine function for specific angles, we will use the same values to plot the points and hence the graph of inverse cosine. For y = cos^{1}x, we have:
 When x = 0, y = π/2
 When x = 1/2, y = π/3
 When x = 1, y = 0
 When x = 1, y = π
 When x = 1/2, y = 2π/3
Cos Inverse x Derivative
Now, we will determine the derivative of inverse cosine function using some trigonometric formulas and identities. Assume y = cos^{1}x ⇒ cos y = x. Differentiate both sides of the equation cos y = x with respect to x using the chain rule.
cos y = x
⇒ d(cos y)/dx = dx/dx
⇒ sin y dy/dx = 1
⇒ dy/dx = 1/sin y  (1)
Since cos^{2}y + sin^{2}y = 1, we have sin y = √(1  cos^{2}y) = √(1  x^{2}) [Because cos y = x]
Substituting sin y = √(1  x^{2}) in (1), we have
dy/dx = 1/√(1  x^{2})
Since x = 1, 1 makes the denominator √(1  x^{2}) equal to 0, and hence the derivative is not defined, therefore x cannot be 1 and 1.
Hence the derivative of cos inverse x is 1/√(1  x^{2}), where 1 < x < 1
Inverse Cosine Integration
We will find the integral of inverse cosine, that is, ∫cos^{1}x dx using the integration by parts (ILATE).
∫cos^{1}x = ∫cos^{1}x · 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^{1}x and g(x) = 1.
∫cos^{1}x · 1 dx = cos^{1}x ∫1 dx  ∫ [d(cos^{1}x)/dx ∫1 dx]dx + C
∫cos^{1}x dx = cos^{1}x . (x)  ∫ [1/√(1  x²)] x dx + C
We will evaluate this integral ∫ [1/√(1  x²)] x dx using substitution method. Assume 1x^{2} = u. Then 2x dx = du (or) x dx = 1/2 du.
∫cos^{1}x dx = x cos^{1}x  ∫(1/√u) (1/2) du + C
= x cos^{1}x  1/2 ∫u^{1/2} du + C
= x cos^{1}x  (1/2) (u^{1/2}/(1/2)) + C
= x cos^{1}x  √u + C
= x cos^{1}x  √(1  x²) + C
Therefore, ∫cos^{1}x dx = x cos^{1}x  √(1  x²) + C
Properties of Inverse Cosine
Some of the properties or formulas of inverse cosine function are given below. These are very helpful in solving the problems related to cos inverse x in trigonometry.
 cos(cos^{1}x) = x only when x ∈ [1, 1] (When x ∉ [1, 1], cos(cos^{1}x) is NOT defined)
 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])
 cos^{1}(x) = π  cos^{1}x
 cos^{1}(1/x) = sec^{1}x, when x ≥ 1
 sin^{1}x + cos^{1}x = π/2, when x ∈ [1, 1]
 d(cos^{1}x)/dx = 1/√(1  x^{2}), 1 < x < 1
 ∫cos^{1}x dx = x cos^{1}x  √(1  x²) + C
Important Notes on Cos Inverse x
 Invere cosine is NOT the same as (cos x)^{1} as (cos x)^{1} = 1/(cos x) = sec x.
 θ = cos^{1}[ (adjacent side) / (hypotenuse) ], θ ∈ [0, π]
 d(cos^{1}x)/dx = 1/√(1  x^{2}), 1 < x < 1
 ∫cos^{1}x dx = x cos^{1}x  √(1  x²) + C
 cos^{1}(x) = π  cos^{1}x
Related Topics on Inverse Cosine
Inverse Cosine Examples

Example 1: Determine the value of x if cos^{1}(√3/2) = x using the inverse cosine formula.
Solution: We have cos x = Adjacent Side/Hypotenuse and cos^{1}x is an inverse function of cos x.
We know that cos (π/6) = √3/2. Since π/6 ∈ [0, π], cos^{1}(√3/2) = π/6
Answer: The value of x if cos^{1}(√3/2) = x using the inverse cosine formula is π/6.

Example 2: Evaluate cos(cos^{1}5) and cos^{1}(cos 5π/3) using inverse cosine properties.
Solution: First we will determine the value of cos(cos^{1}5). Since 5 ∉ [1, 1], cos(cos^{1}5) is not defined.
Now, we will evaluate cos^{1}(cos 5π/3). Since 5π/3 ∉ [0, π], we will determine the equivalent value of 5π/3 that lies in [0, π].
Since cos x = cos (2π  x), we have cos 5π/3 = cos (2π  5π/3) = cos(π/3)
Now, π/3 ∈ [0, π], therefore cos^{1}(cos 5π/3) = cos^{1}(cos π/3) = π/3
Answer: cos(cos^{1}5) is not defined and cos^{1}(cos 5π/3) = π/3
FAQs on Inverse Cosine
What is Inverse Cosine in Trigonometry?
Inverse cosine is the inverse of the cosine function. Inverse cosine of x can also be written as cos^{1}x or arccos x. Then by the definition of inverse cosine, θ = cos^{1}[ (adjacent side) / (hypotenuse) ].
What is Inverse Cosine Formula?
By the definition of inverse cosine, θ = cos^{1}[ (adjacent side) / (hypotenuse) ]. Here θ is the angle between the adjacent side and the hypotenuse and lies between 0 and π.
What is the Derivative of Inverse Cosine?
The derivative of cos inverse x is 1/√(1  x^{2}), where 1 < x < 1. It can be calculated using the chain rule.
What is the Domain and Range of Inverse Cosine?
The domain of the inverse cosine is [1, 1] because the range of the cosine function is [1, 1]. The range of cos inverse x, cos^{1}x^{ }is [0, π]. We need to make the cosine function oneone by restricting its domain R to the principal branch [0, π] which makes the range of the inverse cosine as [0, π].
How to Calculate Integral of Inverse Cosine?
The integral of cos inverse x can be calculated using integration by parts. Integral of inverse cosine is given by ∫cos^{1}x dx = x cos^{1}x  √(1  x²) + C
What Is Cos of Cos Inverse x?
Cos of cos inverse x is x, that is, cos(cos^{1}x) = x if x ∈ [1, 1]. If x ∉ [1, 1] then cos(cos^{1}x) is not defined.
What Is the Inverse Cosine of Cos x?
Inverse cosine of cos x is x, that is, cos^{1}(cos x) = x, if x ∈ [0, π]. If x ∉ [0, π] then apply trigonometric identities to find the equivalent angle of x that lies in [0, π].
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