Inverse Trigonometric Functions

This lesson covers the inverse trigonometric functions — arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹) — their definitions, domains, ranges, graphs, and their use in solving equations. Understanding these functions is essential for solving trigonometric equations and for integration involving inverse trig functions.

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Why Do We Need Inverse Trig Functions?

If sin θ = 0.5, what is θ? You know that θ = 30° is one answer, but θ = 150° also works, as do θ = 390°, θ = −210°, and infinitely many others. The sine function is many-to-one, so it does not have a straightforward inverse unless we restrict its domain.

An inverse function requires the original function to be one-to-one. We achieve this by restricting the domain of sin, cos, and tan to a suitable interval.

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Definitions and Restricted Domains

arcsin (sin⁻¹)

We restrict sin to the domain [−π/2, π/2], where it is one-to-one and covers its full range [−1, 1].

arcsin: [−1, 1] → [−π/2, π/2]

• Domain of arcsin: [−1, 1]
• Range of arcsin: [−π/2, π/2] (i.e., −90° to 90°)