What traumatic event does Zitkala-Sa describe at the school?

The forcible cutting of her long hair

Why was the cutting of her hair so humiliating to Zitkala-Sa?

Among her people, shingled hair was worn by cowards and mourners

State the principal value range of \(\sin^{-1}x\).

The principal range is \([-\tfrac{\pi}{2},\tfrac{\pi}{2}]\), with domain \([-1,1]\).

State the principal value range of \(\cos^{-1}x\).

The principal range is \([0,\pi]\), with domain \([-1,1]\).

Evaluate \(\sin^{-1}\!\left(\tfrac{1}{2}\right)\).

Since \(\sin\tfrac{\pi}{6}=\tfrac{1}{2}\) and \(\tfrac{\pi}{6}\) lies in the principal range, \(\sin^{-1}\!\left(\tfrac{1}{2}\right)=\tfrac{\pi}{6}\).

Prove \(\sin^{-1}x+\cos^{-1}x=\tfrac{\pi}{2}\).

Let \(\sin^{-1}x=\theta\); then \(x=\sin\theta=\cos(\tfrac{\pi}{2}-\theta)\), so \(\cos^{-1}x=\tfrac{\pi}{2}-\theta\), giving the sum \(\tfrac{\pi}{2}\).

Evaluate \(\tan^{-1}(1)\).

\(\tan\tfrac{\pi}{4}=1\) and \(\tfrac{\pi}{4}\) is in the principal range, so \(\tan^{-1}(1)=\tfrac{\pi}{4}\).

Simplify \(\cos^{-1}(\cos\tfrac{7\pi}{6})\).

Since \(\tfrac{7\pi}{6}\notin[0,\pi]\), use \(\cos\tfrac{7\pi}{6}=\cos(2\pi-\tfrac{7\pi}{6})=\cos\tfrac{5\pi}{6}\); thus the value is \(\tfrac{5\pi}{6}\).

State the value of \(\tan^{-1}x+\cot^{-1}x\).

For all real \(x\), \(\tan^{-1}x+\cot^{-1}x=\tfrac{\pi}{2}\).

Write the addition formula for \(\tan^{-1}\).

\(\tan^{-1}x+\tan^{-1}y=\tan^{-1}\dfrac{x+y}{1-xy}\) provided \(xy<1\).

Evaluate \(\sin^{-1}(\sin\tfrac{2\pi}{3})\).

As \(\tfrac{2\pi}{3}\notin[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\), use \(\sin\tfrac{2\pi}{3}=\sin(\pi-\tfrac{2\pi}{3})=\sin\tfrac{\pi}{3}\); hence the value is \(\tfrac{\pi}{3}\).

Why are trig functions restricted before inverting?

They are periodic and not one-one over their full domains; restricting to a principal branch makes them bijective so a single-valued inverse exists.

CBSE · Senior secondary

CBSE Class 12 — Notes, Chapters & Practice Quizzes

Q: Evaluate \(\sin^{-1}(\sin\tfrac{2\pi}{3})\).

As \(\tfrac{2\pi}{3}\notin[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\), use \(\sin\tfrac{2\pi}{3}=\sin(\pi-\tfrac{2\pi}{3})=\sin\tfrac{\pi}{3}\); hence the value is \(\tfrac{\pi}{3}\).

Q: Why are trig functions restricted before inverting?

They are periodic and not one-one over their full domains; restricting to a principal branch makes them bijective so a single-valued inverse exists.

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Chapter 2: Inverse Trigonometric Functions — Class 12 Mathematics

Mathematics · 13 chapters

Summary, key terms, important questions and a practice quiz with AI diagnosis for each.

Chapter 2: Inverse Trigonometric Functions

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The principal value range of \(\sin^{-1}x\) is:

Summary

Trigonometric functions are periodic and therefore not one-one over their full domains. To define inverses, each is restricted to a principal-value branch on which it is bijective. The chapter defines \(\sin^{-1}x,\cos^{-1}x,\tan^{-1}x\) and their reciprocals, listing the principal ranges: \(\sin^{-1}x\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\), \(\cos^{-1}x\in[0,\pi]\) and \(\tan^{-1}x\in(-\tfrac{\pi}{2},\tfrac{\pi}{2})\). Several identities follow, such as \(\sin^{-1}x+\cos^{-1}x=\tfrac{\pi}{2}\) and \(\tan^{-1}x+\tan^{-1}y=\tan^{-1}\dfrac{x+y}{1-xy}\) (for \(xy<1\)). These functions and identities are essential tools for simplifying expressions, solving equations and integrating, where inverse trig functions appear as antiderivatives.

Principal value branchesInverse sine, cosine and tangentProperties and identitiesSimplification of inverse trig expressions

Key terms

Principal value branch

The restricted interval on which a trig function is bijective so its inverse is defined.

\(\sin^{-1}x\)

Inverse sine, with range \([-\tfrac{\pi}{2},\tfrac{\pi}{2}]\) and domain \([-1,1]\).

\(\cos^{-1}x\)

Inverse cosine, with range \([0,\pi]\) and domain \([-1,1]\).

\(\tan^{-1}x\)

Inverse tangent, with range \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\) and domain \(\mathbb{R}\).

Complementary identity

\(\sin^{-1}x+\cos^{-1}x=\tfrac{\pi}{2}\).

Addition formula

\(\tan^{-1}x+\tan^{-1}y=\tan^{-1}\dfrac{x+y}{1-xy}\) when \(xy<1\).

Important questions

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Principal value branch

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Meaning1 / 6

The restricted interval on which a trig function is bijective so its inverse is defined.

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Practice quiz · Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Maths 10 Qs · ~10 min