Circular Representation of Inverse Trigonometric Functions

Consider a function f. The inverse of this function f exists only if f is one to one function. It is given by

f(x)=y and f-1(y)=x

For the sine function sin: R→R. For all θ € R the sinθ = x is a one-to-one function. Here for the domain R the sine function doesn’t have an inverse. But when the domain is considered in [-π/2, π/2] there are infinite values for various angles of θ. In this domain, the equation sinθ = x can be satisfied. From the equation it can be written that θ = sin-1(x).

Read Also: Inverse Trigonometric Formulas

Key Terms: Single-valued function, multiple-valued function, one to one function, sine function, properties of inverse trigonometric function

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Circular Representation of Inverse Trigonometric Functions

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The most important thing to be noted here is that is θ is an angle and it represents the value sin-1(x). In short, θ is the angle and x is the value of the arcsine function of the angle. For different angles, the arcsine function has different values ranging from -1 to 1 and the angle between -π/2 and π/2. So, sin-1(x) is a multiple-valued function.

Read More: Class 10 Introduction to Trigonometry

But the sine function has a certain value for a certain angle. For every sin θ there is a value x. So, sinθ is a single-valued function. All real numbers are the domain of sine function and the range is [-1,1].

sin^-1(x)= θ

x=sin θ

Where, -π/2 ≤ x ≤ π/2 and -1 ≤ x ≤ 1

The sine function is considered as one to one function during the intervals -π/2 and π/2. But other intervals like [3π/2,5π/2], [-5π/2, -3π/2] also define the inverse of sine function. Hence sin−1x is a function with domain [-1, 1] = {x ∈ R: - 1 ≤ x ≤ 1} and range [- π², π²] or [3π², 5π²] or [- 5π², -3π²] and etc.,

Read More: Trigonometric Identities

In the same way, cos θ=x(-1≤x≤1) then θ=cos^-1(x). Here θ is the angle and the value of cosine function of this angle is x. For all the tan^-1x, cot^-1x, sec^-1x, cosec^-1x there is an angle θ and the value of inverse functions of these angles is x.

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Properties

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• If sin θ=x (-1≤x≤1) then θ=sin^-1(x).
• If cos θ=x (-1≤x≤1) then θ=cos^-1(x).
• If tan θ=x (-∞<x </x) then θ=tan^-1x).
• If cot θ=x (-∞<x </x) then θ=cot^-1(x).
• If cosec θ=x (|x|≥ 1) then θ=cosec^-1(x).
• If sec θ=x (|x|≥ 1) then θ=sec^-1(x)

Vice-versa

• sin^-1(x)=θ then sin θ=x
• cos^-1(x)= θ then cos θ=x
• tan^-1(x)= θ then tan θ=x
• cot^-1(x)= θ then cot θ=x
• cosec^-1(x)= θ then cosec θ=x
• sec^-1(x)= θ then sec θ=x

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