Inverse trigonometry

What is inverse trigonometry?

Inverse trigonometric functions or arcus functions are obtained by restricting the six trigonometric ratios which include the sine, cosine, secant, cosecant, cotangent, and tangent functions. This results in both the codomain and the image of the functions appearing as the domain subsets of their parent functions. Inverse trigonometry finds its application in geometry, physics, and engineering. 

The arcus functions execute the reverse operation of the original trigonometric functions. ‘Arc’ is used in front of any trigonometric functions to denote their inverse functions. For example:  arcsin (z), arctan (z) and arcsec (z). 

Here are a couple of principles that are to be remembered while working with inverse trigonometric functions:

1. If we encounter two different angles having identical values then we must consider the angle with a positive value over the negative one. 
2. In inverse trigonometry, any angle or real number whose sin is z is represented by sin-1z. Similarly, the angles whose cosine and tan are z, are marked as cos-1z and tan-1z. 

Inverse trigonometry – Range and Domain

We are about to discuss the range and domain of the six principal trigonometric functions. 

For the function sin θ, we notice identical values on the map for multiple angles that are denoted by θ. 

sin 0 = 0 = sin π = sin 2 π = … = sin c π. 

In the above example, c stands for any integer. Thus, the domain of the trigonometric functions needs to be restricted to determine their arc functions. 

The domain of x for the function y = sin-1x is given by – 1 ≤ x ≤ 1. 

The range for y = – π/2 ≤ y ≤ π/2

To draw the graph of inverse trigonometric functions, we should flip the graph of their original functions about the y = xy = x line. 

The domain of arccosine x or cos-¹x is – 1 ≤ x ≤ 1. 

Thus the domain of arccosine x is similar to sin-¹x.

Range of cos-¹x 0 ≤ y ≤ π. 

The domain of the arccosecant x or cosec-1x function is -∞ ≤ x ≤ -1 

The respective range is 0 ≤ y ≤ π. 

Although π/2 is an intermediary value, it cannot be a possible value of y = cosec-1x. 

The domain of arcsec x is similar to arccosecant x. 

The range of sec-¹x is 0 ≤ y ≤ π. 

The domain of the arctangent x function or tan-¹x is -∞ ≤ x ≤ ∞. 

Therefore the value of x can be any real number.

The range of tan-¹x is – π/2 ≤ x ≤ π/2. 

The domain of cot-1x function matches that of tan-¹x. 

The range of arccotangent x is 0 < y < π. 

Inverse Trigonometry Solved Examples 

1.Determine the numerical value of sin (cos-¹ 3/5).

Ans. Let us assume the angle cos-¹3/5 as α. 

Therefore, according to the principles of inverse trigonometry cos α= 3/5

We know that sin α = √(1 – cos² α) 

⇒ sin α = √(1 – 9/25)

= 16/25

= 4/5

∴ sin (cos-¹ 3/5) = 4/5.

2.What is the value of sec cosec-¹ (2/3)?

Ans. By studying the domain of cosec, we know that cosec π/3 = 2/3

Therefore, sec cosec-¹(2/3) = sec cosec-¹ (cosec π/3) 

Or, sec cosec-¹(2/3) = sec (cosec-¹ cosec π/3) = sec π/3

∴ sec cosec-¹ (2/3) = 2

3.Determine the numerical value both in degrees and in radian for sin-¹(-1/2). 

Ans. Let us assume the angle sin-¹ (-1/2) as α. 

∴ sin α = – 1/2

Range of arcsine α – π/2 ≤ α ≤ π/2. 

So, the value of α must be somewhere within the range to satisfy the condition sin α = – 1/2

Hence, α = –  π/6. This is the radian value of sin-¹(-1/2). 

The degree value is 180°/6 = 30°.

4.Determine the radian value for cos-¹ (√3 /2).

Ans. We can represent the angle cos-¹ (√3 /2) as β. 

∴ cos β = √3 /2

Range of cosine  β 0 ≤  β ≤ π

So, the value of β must be somewhere within the range to satisfy the condition cos β = √3 /2

Hence, β =  π/6. This is the radian value of cos-¹ (√3 /2). 

5.Determine the domain of cos-¹( [2 + sin x] /3).

Let the inverse function of arccosine x be replaced by α. 

Then by the definition of the domain, |α|≤ 1 as cos-¹ α ranges between – 1 and 1.

This concludes that

-1 ≤ [2 + sin x] /3 ≤ 1

= – 3 ≤ 2 + sin x ≤ 3 ⇒ – 5  ≤ sin x ≤ 1

When we reduce the equation we get – 1 ≤ sin x ≤ 1

This can alternatively be written as – sin-¹ (1) ≤ x ≤ sin-¹ (1)

Thus the value of x lies between – π/2 and π/2. 

∴ Cos-¹ ( [2 + sin x] /3) has a domain between [-  π/2, π/2]. 

Conclusion 

The reverse functionality of regular trigonometric ratios is displayed by inverse trigonometric functions. A minus 1 subscript is used over the concerned trigonometric function to represent its inverse counterpart. Inverse trigonometry is used in geometry, physics, and engineering.