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Arccosine is one of the inverse trigonometric functions and it is also written as cos-1. Since cos-1(x) is the inverse of cos(x), arccosine (x) is the inverse function of cos x. We have 6 inverse trigonometric functions such as
- arcsin = inverse of sin = sin-1
- arccos = inverse of cos = cos-1
- arctan = inverse of tan = tan-1
- arccsc = inverse of csc = csc-1
- arcsec = inverse of sec = sec-1
- arccot = inverse of cot = cot-1
Here, we will study in detail about the inverse cos function (arccosine) along with its graph, domain, range, formulas, derivative, and integral along with a few solved examples.
|1.||What Is Arccosine?|
|2.||Domain, Range, and Graph of Arccosine|
|3.||Properties of Arccosine|
|4.||Derivative of Arccos x|
|5.||Integral of Arccos x|
|6.||FAQs on Arccosine|
What is Arccosine?
Arccosine is the inverse of the cosine function and thus it is one of the inverse trigonometric functions. Arccosine is pronounced as "arc cosine". Arccosine of x can also be written as "acosx" (or) "cos-1x" or "arccos". If f and f-1 are inverse functions of each other, then f(x) = y ⇒ x = f-1(y). So y = cos x ⇒ x = cos-1(y). This is the meaning of arccosine. Let us consider a few examples to see how the arccosine function works.
- cos 0 = 1 ⇒ 0 = arccos (1)
- cos π/2 = 0 ⇒ π/2 = arccos (0)
- cos π = -1 ⇒ π = arccos (-1)
In a right-angled triangle, the cosine of an angle (θ) is the ratio of its adjacent side to the hypotenuse. i.e., cos θ = (adjacent side) / (hypotenuse). Then by the definition of arccosine, θ = cos-1[ (adjacent side) / (hypotenuse) ] .
Thus, the arccosine function is used to find the unknown angles in a right-angled triangle. Also, it can be used to find the unknown angles in any triangle by using the law of cosines. For examples, in a triangle ABC, if AB = c, BC = a, and CA = b, then by the law of cosines,
a2 = b2 + c2 - 2bc cos A
cos A = (b2 + c2 - a2) / (2bc)
A = cos-1[(b2 + c2 - a2) / (2bc)] (or) arccosine[(b2 + c2 - a2) / (2bc)].
Similarly, we can find the other angles of the triangle given its side lengths.
Domain, Range, and Graph of Arccosine
In this section, let us see how can find the domain and range of the arccosine function. Also, we will see how to graph it in its principal domain.
Domain and Range of Arccosine
We know that the cosine function is a function from R → [-1, 1]. But cosine function is NOT a bijection (as it is NOT one-one) on the domain R. Hence it cannot have an inverse if its domain is R. For the cosine function to be one-one, its domain can be restricted to one of the intervals [-π, 0], [0, π], [π, 2π], etc. Corresponding to each of these intervals, we get a branch of arccosine. The branch of arccosine with the range [0, π] is called the principal branch. Thus, the domain of cosine is usually restricted to be [0, π] and its range is [-1, 1].
We know that the domain and range of a function will be the range and domain of its inverse function respectively. Hence, the domain of the inverse of cosine, which is arccosine, is [-1, 1] and its range is [0, π]. i.e.,
arccos x (or) cos-1x : [-1, 1] → [0, π]
Graph of Arccosine
The graph of the arccosine function with its range to be principal branch [0, π] can be drawn using the following table. Here, we have chosen random values for x in the domain of arccosine which is [-1, 1].
|x||y = cos-1x (or) arccos x|
|-1||cos-1(-1) = π - 0 = π|
|-0.5||cos-1(-0.5) = π - π/3 = 2π/3|
|0||cos-1(0) = π/2|
|0.5||cos-1(0.5) = π/3|
|1||cos-1(1) = 0|
By plotting these points on the graph, we get arccos graph.
Properties of Arccosine
Here are some properties/formulas of arccosine. These are very helpful in solving the problems related to inverse cos in trigonometry.
- cos(cos-1x) = x only when x ∈ [-1, 1]
([When x ∉ [-1, 1], cos(cos-1x) is NOT defined)
- cos-1(cos x) = x, only when x ∈ [0, π]
(When x ∉ [0, π], either find the coterminal angle of x, or apply trigonometric identities to find the equivalent angle of x that lies in [0, π] )
- cos-1(-x) = π - cos-1x
- cos-1(1/x) = sec-1x, when |x| ≥ 1
- sin-1x + cos-1x = π/2, when x ∈ [-1, 1]
Derivative of Arccos x
Let us find the derivative of y = cos-1x. By the definition of arccosine, y = cos-1x can be written as cos y = x. Differentiating this on both sides with respect to x using the chain rule,
- sin y (dy/dx) = 1
dy/dx = -1/sin y ... (1)
Now, we have sin2y + cos2y = 1 ⇒ sin2y = 1 - cos2y ⇒ sin y = √(1 - cos²y) = √1 - x².
Substituting this in (1),
dy/dx = -1/√1 - x²
Thus, the arccosine derivative (or) the derivative of cos-1x is -1/√(1 - x²).
Integral of Arccos x
We will find ∫cos-1x dx using the integration by parts. For this, we write the above integral as
∫cos-1x · 1 dx
Using LIATE, f(x) = cos-1x and g(x) = 1.
By integration by parts,
∫f(x) . g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) ∫g(x) dx) dx + C
∫cos-1x · 1 dx = cos-1x ∫1 dx - ∫ [d/dx(cos-1x) ∫x dx] + C
∫cos-1x dx = cos-1x (x) - ∫ [-1/√1 - x²] x dx + C
We will evaluate this integral using u-substitution. For this, let 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.
Important Notes on Arccosine:
Here are some important points to note related to the arccosine function.
- arccosine can be written as cos-1 (or) arccos (or) acos and it is a function with domain [-1, 1] and range [0, π].
- arccosine is NOT same as (cos x)-1 as (cos x)-1 = 1/(cos x) = sec x.
- cos-1(-x) is NOT -cos-1(x), rather cos-1(-x) = π - cos-1x.
- cos(cos-1x) is NOT always x. cos(cos-1x) = x only when x ∈ [-1, 1].
- cos-1(cos x) is NOT always x. cos-1(cos x) = x only when x ∈ [0, π].
Here are some topics that you might be interested in while reading about arccosine.
Solved Examples Using Arccosine
Example 1: If θ is an acute angle in a right triangle whose adjacent side is 2 units and the hypotenuse is 4 units, find θ.
We know that cos θ = (adjacent side) / (hypotenuse) = 2/4 = 1/2.
We know that arccosine (which is written as cos-1) is the inverse of cos. So
θ = cos-1 (1/2) = π/3.
(This is because from the trigonometric table, cos π/3 = 1/2)
Answer: θ = π/3.
Example 2: Find the values of the following: a) cos(cos-1 2) b) cos-1(cos 7π/6).
a) We know that cos(cos-1 x) is NOT defined when x is NOT in [-1, 1] as the domain of arccosine is [-1, 1].
Thus, cos(cos-1 2) is NOT defined.
b) cos-1(cos 7π/6)
Please note that this is NOT equal to 7π/6 as 7π/6 ∉ [0, π].
So we have to convert this angle to lie in the interval [0, π].
Since we have cos x = cos (2π - x),
cos 7π/6 = cos(2π- 7π/6) = cos(5π/6)
We have 5π/6 ∈ [0, π].
Thus, cos-1(cos 7π/6) = cos-1(cos 5π/6) = 5π/6
Answer: a) cos(cos-1 2) is NOT defined b) cos-1(cos 7π/6) = 5π/6.
Example 3: Prove that sin-1(2x √1 - x²) = 2 cos-1x, 1/√2 ≤ x ≤ 1.
LHS = sin-1(2x √1 - x²)
Substitute x = cos θ. Then by the definition of arccosine, θ = cos-1(x) ... (1)
Now, substitute x = cos θ in the LHS,
= sin-1(2cos θ √1 - cos²θ)
= sin-1(2cos θ √sin²θ)
= sin-1(2cos θ sin θ)
= sin-1(sin 2θ)
= 2 cos-1(x) (from (1))
Answer: We have proved sin-1(2x √1 - x²) = 2 cos-1x.
Practice Questions on Arccosine
FAQs on Arccosine
What is Arccosine Meaning?
Arccosine is the inverse function of the trigonometric function cos x and hence it is an inverse trigonometric function. By the definition of inverse function, y = cos x ⇒ x = cos-1(y).
What is Arccosine Formula?
In a right triangle, if θ is one of the acute angles, then cos θ = (adjacent)/(hypotenuse). Then θ = arccos((adjacent)/(hypotenuse). This is the formula of arccosine (or arccos).
Is Arccosine of x Same As cos⁻¹x?
Arccosine is the inverse of cos x and so yes, arccosine of x is the same as cos⁻¹x.
Is Arccosine Same As (cos x)⁻¹?
No. Arccosine is the inverse of cosine function, i.e., arccos x = cos-1x. But it is NOT equal to (cos x)-1, as (cos x)-1 = 1/ cos x = sec x.
What is Arccosine Pronunciation?
"Arccosine" is one of the inverse trigonometric functions and it can be pronounced as "arc cosine" or "inverse cosine".
What is the Domain of Arccos x?
The domain of arccosine (or) arccos x is [-1, 1] as the range of its inverse (which is cosine function) is [-1, 1].
What is the Range of Arccos x?
The range of arccos x (or) cos-1x is [0, π] as the restricted domain of its inverse (which is the cosine function) is [0, π] to make the cosine function one-one.
What is Arccosine of Cos x?
Arccosine of cos x is x (or) cos-1(cos x) = x, if x ∈ [0, π]. If x ∉ [0, π] then either find the coterminal angle of x, or apply trigonometric identities to find the equivalent angle of x that lies in [0, π] and then apply cos-1(cos x) = x.
What is Cos of Arccosine of x?
Cos of arccosine of x is x (or) cos(cos-1x) = x if x ∈ [-1, 1]. If x ∉ [-1, 1] then cos(cos-1x) is NOT defined.