Cos Inverse Formula

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Cos Inverse Formula

Cos Inverse is also written as Arccos and Cos-1 and is called an anti trigonometric function. The cos inverse is an inverse trigonometric function with restricted domains.

The Cos inverse Formula is :

Cos x = Adjacent / Hypotenuse
Cos-1 (Adjacent/Hypotenuse) = x

Arc cos Questions

Cos inverse formula is one amongst the other six inverse trigonometric functions.

Question: Find the exact Value of Arccos(-½)

Solution:

arccos(- 1 / 2)

Assume y = arccos(- 1 / 2).

cos y = – 1 / 2 with 0 ≤ y ≤ Π (Cos Theorem) … (1)

Cos (π / 3) = 1 / 2 (Use table of Special Angles)

And also, cos(Π – x) = – cos x.

cos (π – π/3) = – 1 / 2 …(2)

Compare statement 1 with statement 1

y = Π – Π / 3 = 2 Π /3

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-1x, cos-1x, tan-1x, csc-1x, sec-1x, cot-1x.

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 right-angled 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 right-angled 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(one-one and onto). Since cos x is not a bijective function as it is not one-one, the inverse cosine cannot have R as its range. Hence, we need to make the cosine function one-one 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-1x, 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-1x ⇒ 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 cos2y + sin2y = 1, we have sin y = √(1 – cos2y) = √(1 – x2) [Because cos y = x]

Substituting sin y = √(1 – x2) in (1), we have

dy/dx = -1/√(1 – x2)

Since x = -1, 1 makes the denominator √(1 – x2) 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 – 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 (ILATE).

∫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

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-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], apply the 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]
  • d(cos-1x)/dx = -1/√(1 – x2), -1 < x < 1
  • ∫cos-1x dx = x cos-1x – √(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-1x)/dx = -1/√(1 – x2), -1 < x < 1
  • ∫cos-1x dx = x cos-1x – √(1 – x²) + C
  • cos-1(-x) = π – cos-1x

Inverse Cosine Examples

What Is Cos Inverse Formula?

Cos inverse formula is used to find the angle when base and hypotenuse are given. Cos inverse function also possesses some properties regarding its domain and range. The Cos inverse function is also called as “Arc Function”. Cos inverse formula can be given as:

Examples Using Cos Inverse Formula

Example 1: In a right-angled triangle ABC, the right angle at B, base AB = 12 inches, and AC = 24 inches. Find angle A in the triangle.

Solution:

To Find: Angle A.
Given:
Base = 12 inch
Hypotenuse = 24 inch
Now, using the cos inverse formula:

Answer: Angle A will be 60∘60∘.

Example 2: A ladder 48.5 inches in length is used to climb a 42-inch tall building. Find the angle made by the ladder with the wall of the building to reach its top.

Solution:

To Find : Angle made by the ladder with the wall of the building.
Given:
Base = 42 inch
Hypotenuse = 48.5 inch
Now, using the cos inverse formula:

Inverse Trigonometry Formula

What is an Inverse Trigonometric Function?

In geometry, the part that tells us about the relationships existing between the angles and sides of a right-angled triangle is known as trigonometry. It has formulas and identities that offer great help in mathematical and scientific calculations. Along with that trigonometry also has functions and ratios such as sin, cos, and tan. In the same way, we can answer the question of what is an inverse trigonometric function? 

Well, there are inverse trigonometry concepts and functions that are useful. Inverse trigonometry formulas can help you solve any related questions. Just as addition is an inverse of subtraction and multiplication is an inverse of division, in the same way, inverse functions in an inverse trigonometric function. We can call it by different names such as anti-trigonometric functions, arcus functions, and cyclometric functions. The inverse trigonometric functions are as popular as anti trigonometric functions. The inverse functions have the same name as functions but with a prefix “arc” so the inverse of sine will be arcsine, the inverse of cosine will be arccosine, and tangent will be arctangent. So now when next time someone asks you what is an inverse trigonometric function?  You have a lot to say.  

y=sin-1x

Inverse Trigonometric Functions Formulas

Inverse trigonometric functions formula helps the students to solve the toughest problem easily, all thanks to inverse trigonometry formula. Some of the inverse trigonometric functions formulas are:

  1. sin-1(x) = – sin-1x
  2. cos-1(x) = π – cos-1x
  3. sin-1(x) + cos-1x = π/2

So these were some of the inverse trigonometric functions formulas that you can use while solving trigonometric problems

Table of Inverse Trigonometric Functions

Fun Facts

  • Hipparchus, the father of trigonometry compiled the first trigonometry table
  • Inverse trigonometric functions were actually introduced early in 1700x by Daniel Bernoulli. 

Solved Examples 

Example 1) Find the value of tan-1(tan 9π/ 8 )

Solution 1)  tan-1(tan9π/8)       

= tan-1tan ( π + π/8)

 = tan-1 (tan(π/ 8))

=π / 8

Example 2) Find sin (cos-13/5)

Solution 2) Suppose that, cos-1 3/5 = x

So, cos x = 3/5

This implies, sin x = sin (cos-1 3/5) = ⅘

Example 3) Prove the equation “Sin-1 (-x) = – Sin-1 (x), x ϵ (-1, 1)”

Solution 3) Let Sin-1 (-x) = y

Then -x = sin y

x = – sin y

x =sin (-y)

sin-1 -x = arcsin ( sin(-y))

sin-1 -x = y

Hence, Sin-1 (-x) = – Sin-1 (x), x ϵ (-1, 1)

Example 4) Prove – Cos-1 (4x3 -3 x) =3 Cos-1 x , ½ ≤ x ≤ 1

Solution 4)  Let x = Cos ϴ

Where ϴ = Cos-1 (-x)

LHS = Cos-1 (-x) (4x3 -3x)

By substituting the value of x, we get

= Cos-1 (-x) (4 Cos3ϴ – 3 Cosϴ)

Accordingly, we get,

 Cos-1 (Cos 3ϴ)

= 3ϴ

By substituting the value of ϴ, we get

= 3 Cos-1 x

= RHS

Hence proved

FAQs (Frequently Asked Questions)

Question 1) What are the applications of Inverse Trigonometric Functions?

Answer 1) The inverse trigonometric formula’s major role is to help us in finding out the unknown measurement of an angle of a right angle triangle when any of its two sides are provided. We use the trigonometric function particularly on the basis of which sides are known to us. In other words, if the measurement of the side of the hypotenuse and the side opposite to the angle ϴ are known to us then we use an inverse sine function. In the same way, if we are provided with the measurement of the adjacent side and the opposite side then we use an inverse tangent function for the determination of a right-angle triangle. The inverse trigonometric function extends its hand even to the field of engineering, physics, geometry, and navigation. An inverse trigonometric function can be determined by two methods. The first is to use the trigonometric ratio table and the second is to use scientific calculators.

Question 2) What are Trigonometric Functions?

Answer 2) Trigonometry is the science of measuring triangles. We can refer to trigonometric functions as the functions of an angle of a triangle. In other words, it is these trig functions that define the relationship that exists between the angles and sides of a triangle. 

There are six main trigonometric functions that are given below:

  1. Sine (sin)
  2. Cosine (cos)
  3. Tangent (tan)
  4. Secant (sec)
  5. Cosecant (cosec)
  6. Cotangent (cot)

We use these functions to relate the angles and the sides of a right-angled triangle. Trigonometric functions are important when we are studying triangles. 

Inverse Trigonometric Formulas

Inverse Trigonometric Formulas: Trigonometry is a part of geometry, where we learn about the relationships between angles and sides of a right-angled triangle. In Class 11 and 12 Maths syllabus, you will come across a list of trigonometry formulas, based on the functions and ratios such as, sin, cos and tan. Similarly, we have learned about inverse trigonometry concepts also. The inverse trigonometric functions are written as sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, sec-1 x. Now, let us get the formulas related to these functions.

What is Inverse Trigonometric Function?

The inverse trigonometric functions are also known as the anti trigonometric functions or sometimes called arcus functions or cyclometric functions. The inverse trigonometric functions of sine, cosine, tangent, cosecant, secant and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics, etc. But in most of the time, the convention symbol to represent the inverse trigonometric function using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). To determine the sides of a triangle when the remaining side lengths are known.

Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f-1,

This means that if y=f(x), then x = f-1(y).

Such that f(g(y))=y and g(f(y))=x.

Example of Inverse trigonometric functions: x= sin-1y

The list of inverse trigonometric functions with domain and range value is given below:

FunctionsDomainRange
Sin-1 x[-1, 1][-π/2, π/2]
Cos-1x[-1, 1][0, π/2]
Tan-1 xR(-π/2, π/2)
Cosec-1 xR-(-1,1)[-π/2, π/2]
Sec-1 xR-(-1,1)[0,π]-{ π/2}
Cot-1 xR[-π/2, π/2]-{0}

Inverse Trigonometric Formulas List

To solve the different types of inverse trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. The formula list is given below for reference to solve the problems.

S.NoInverse Trigonometric Formulas
1sin-1(-x) = -sin-1(x), x ∈ [-1, 1]
2cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]
3tan-1(-x) = -tan-1(x), x ∈ R
4cosec-1(-x) = -cosec-1(x), |x| ≥ 1
5sec-1(-x) = π -sec-1(x), |x| ≥ 1
6cot-1(-x) = π – cot-1(x), x ∈ R
7sin-1x + cos-1x = π/2 , x ∈ [-1, 1]
8tan-1x + cot-1x = π/2 , x ∈ R
9sec-1x + cosec-1x = π/2 ,|x| ≥ 1
10sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1
11cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1
12tan-1(1/x) = cot-1(x), x > 0
13tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1
14tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1
152 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1
162tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0
172tan-1 x = tan-1(2x/(1-x2)), -1<x<1
183sin-1x = sin-1(3x-4x3)
193cos-1x = cos-1(4x3-3x)
203tan-1x = tan-1((3x-x3)/(1-3x2))
21sin(sin-1(x)) = x, -1≤ x ≤1
22cos(cos-1(x)) = x, -1≤ x ≤1
23tan(tan-1(x)) = x, – ∞ < x < ∞.
24cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞
25sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞
26cot(cot-1(x)) = x, – ∞ < x < ∞.
27sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2
28cos-1(cos θ) = θ, 0 ≤ θ ≤ π
29tan-1(tan θ) = θ, -π/2 < θ < π/2
30cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2
31sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π
32cot-1(cot θ) = θ, 0 < θ < π

The inverse trigonometric formula list helps the students to solve the problems in an easy way by applying those properties to find out the solutions.

Inverse Sine, Cosine, Tangent

Inverse Sine Function

But sometimes it is the angle we need to find.

This is where “Inverse Sine” comes in.

It answers the question “what angle has sine equal to opposite/hypotenuse?”

The symbol for inverse sine is sin-1, or sometimes arcsin.

They Are Like Forward and Backwards!

The Graphs

And lastly, here are the graphs of Sine, Inverse Sine, Cosine and Inverse Cosine:

Did you notice anything about the graphs?

  • They look similar somehow, right?
  • But the Inverse Sine and Inverse Cosine don’t “go on forever” like Sine and Cosine do …

But we saw earlier that there are infinitely many answers, and the dotted line on the graph shows this.

So yes there are infinitely many answers …

… but imagine you type 0.5 into your calculator, press cos-1 and it gives you a never ending list of possible answers …

So we have this rule that a function can only give one answer.

So, by chopping it off like that we get just one answer, but we should remember that there could be other answers.

Differentiation of Inverse Trigonometric Functions

Each of the six basic trigonometric functions have corresponding inverse functions when appropriate restrictions are placed on the domain of the original functions. All the inverse trigonometric functions have derivatives, which are summarized as follows:

Related articles

Cosecant Formula

Cos Double Angle Formula

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Cos Inverse Formula ⭐️⭐️⭐️⭐️⭐

Cos Theta Formula ⭐️⭐️⭐️⭐️⭐

Cosec Cot Formula ⭐️⭐️⭐️⭐️⭐

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2cosacosb Formula

Law of Cosines Formula ⭐️⭐️⭐️⭐️⭐

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