What is the difference between y = arccos x and y = cos-1 x?

Trigonometry is the branch of Mathematics that establishes a relation between sides and the angles of a right-angled triangle. It helps to find the unknown sides of the unknown angles of the triangle. The angles are measured either in radians or in degrees. The trigonometric ratios or trigonometric functions are sine, cosine, tangent, and their inverses sec, cosec, and cot. The trigonometric functions are calculated using a right-angled triangle. The longest side in the triangle is the hypotenuse and the sides opposite are the base and height. Trigonometric functions are used in obtaining unknown angles and distances.

1. Sine is the ratio of base to the hypotenuse
2. Cosine is the ratio of height to the hypotenuse
3. Tangent is the ratio of height to base

Inverse of Trigonometric Functions

The inverse of Trigonometric functions is used to obtain angles from the ratios. They are also known as ‘Arc functions’. The inverse is applicable on sine, cosine, tangent, cosecant, secant, and cotangent. In this domain and range also gets inverted. It is represented by cos^-1, sin^-1,tan^-1 etc. They are often represented by arccos, arcsin, arctan, etc.  The inverse function is also known as the anti trigonometric function. These functions are useful in engineering, physics, mathematics, etc. Some formulas are mentioned below,

1. cos^-1x + sin^-1x = π/2 
2. arctan(x) + arc cot(x) = π/2
3. arcsec(x) + arc cosec(x) = π/2

What is the difference between y = arccos x and y = cos^-1 x?

Answer:

As we all know cos of any angle is defined as the ratio of base to the hypotenuse. cos^-1 is basically the inverse of cos x. cos inverse is denoted by cos^-1 (Base/Hypotenuse). It should be noted inverse of cosine is not the reciprocal of cosine. The inverse of this function is also known as arc cosine or written as acos. This function returns the angle based on value. The domain is [-1,1] and range is [0, π]. Therefore acos and cos^-1 are the same. Therefore the definition of cos^-1 or acos is given by,

Let y = cos x

Therefore cos^-1 (y) = acos (y) = x

Example: Find the cos^-1(√3/2)

cos x= √3/2

cos x = cos π/6

x = π/6

The other formulas,

• arc cos(1/x) = sec^-1(x)
• cos^-1(-x) = π -cos^-1(x)
• cos(cos^-1 x) = x
• cos^-1 x + cos^-1 y= cos^-1(xy – √(1 – x²)√(1 – y² ))
• cos^-1 x – cos^-1 y= cos^-1(xy + √(1-x²)√(1-y²))

Similar Questions

Question 1: Find the arc cosine if 

1. cos x = 0
2. cos x = 1

Solution:

1. Let y = cos x = 0

cos x= 0

x = arc cos(0)

x = π/2

2. Let y = cos x = 1

cos x= 1

x = arc cos(1)

x = 0

Question 2: Find arc cos x if sinx is 0.5

Solution:

As we all know arccos(x) + arcsin(x) = π/2

Therefore arc cos(x) = π/2 – arc sin(0.5)

arccos(x) = π/3

Question 3: Find the domain and the range of arccos(x). Hence find the value of cos^-1(2) 

Solution:

The range of arccos(x) is [0,π] and the domain of arccos(x) is [-1,1]. 

Since the domain of inverse cosine is [-1,1] therefore the inverse does not exist

Question 4: Find the value of cos(cos^-1 0.5) and find the value of cos^-1(-1/√2)

Solution:

As we all know cos(cos^-1 x) = x

Therefore the value is 0.5

As we all know the range is [0,π] and cos^-1(-x) = π -cos^-1(x)

cos^-1(1/√2) =  π/4

The value is  π – π/4 = 3π/4 which lies in the range [0,π]

Question 5: Find the value of cos^-1 (3/5) + cos^-1(7/25) in terms of cos.

Solution:

Let A = cos-1 (3/5) 

B= cos-1 (7/25)

Let

A + B = C 

 => cos( A + B ) = cos C

=> cosAcosB – sinAsinB = cos C

=> 3/5 × 7/25 – 4/5 × 24/25 = cos C

=>cos C = -75/125

=>cos C = -3/5

Question 6: Find value of cos^-1 x – cos^-1 y if x = 3/5 y = 4/5

Solution: 

Given, x = 3/5 y = 4/5

As we know cos^-1 x – cos^-1 y = cos ^-1 (xy + √(1 – x²) √(1 – y²))

cos^-1( 0.6 × 0.8 + 0.6 × 0.8) = cos ^-1 ( 0.96 )