Mathematics

Problem

(a) Define the term “principal value” of an inverse trigonometric function. (b) State the principal value ranges of

\arcsin x

\arccos x

and

\arctan x

. (c) For each of the following expressions, decide whether the angle shown is the principal value of the inverse trigonometric expression indicated. Give a brief reason for every answer. (i)

\arcsin\left(\dfrac12\right)=\dfrac{5\pi}{6}

(ii)

\arccos\left(-\dfrac34\right)=\dfrac{2\pi}{3}

(iii)

\arctan(2)=\dfrac{5\pi}{6}

(d) Rewrite every incorrect equality in part (c) with the correct principal value.

Intuitive explanation

Inverse trigonometric functions are multi-valued because many angles produce the same sine, cosine, or tangent—so we restrict them to a single, agreed-upon range called the principal value range. This restriction ensures each input has exactly one output, making $\arcsin$ , $\arccos$ , and $\arctan$ true functions. To check whether a given angle is the principal value, you verify both that the trigonometric equation is true and that the angle lies within the designated range for that function.

Intuitive explanation

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