What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Convert \(240^\circ\) into radian measure.

Radian \(=\dfrac{\pi}{180}\times 240=\dfrac{4\pi}{3}\) radians.

State the relation between arc length, radius and angle.

If an arc of length \(l\) subtends an angle \(\theta\) (radians) at the centre of a circle of radius \(r\), then \(l=r\theta\).

Give the range and period of \(\sin x\) and \(\cos x\).

Both have domain \(\mathbb{R}\), range \([-1,1]\) and period \(2\pi\).

Find \(\sin 15^\circ\) using the difference formula.

\(\sin 15^\circ=\sin(45^\circ-30^\circ)=\dfrac{\sqrt3-1}{2\sqrt2}\).

Write the three double-angle forms of \(\cos 2x\).

\(\cos 2x=\cos^2 x-\sin^2 x=2\cos^2 x-1=1-2\sin^2 x\).

Express \(\tan(x+y)\) in terms of \(\tan x\) and \(\tan y\).

\(\tan(x+y)=\dfrac{\tan x+\tan y}{1-\tan x\tan y}\), provided none of \(x,y,(x+y)\) is an odd multiple of \(\tfrac{\pi}{2}\).

When does \(\sin x=0\) and when does \(\cos x=0\)?

\(\sin x=0\) when \(x=n\pi\); \(\cos x=0\) when \(x=(2n+1)\tfrac{\pi}{2}\), \(n\in\mathbb{Z}\).

State the sum-to-product formula for \(\sin x+\sin y\).

\(\sin x+\sin y=2\sin\dfrac{x+y}{2}\cos\dfrac{x-y}{2}\).

Give the formula for \(\sin 3x\) and \(\cos 3x\).

\(\sin 3x=3\sin x-4\sin^3 x\) and \(\cos 3x=4\cos^3 x-3\cos x\).

In which quadrants are \(\sin x\) and \(\cos x\) positive?

\(\sin x\) is positive in the first and second quadrants; \(\cos x\) is positive in the first and fourth quadrants.

CBSE · Senior secondary

CBSE Class 11 — Notes, Chapters & Practice Quizzes

Master every Class 11 chapter — the base your board marks and your NEET / JEE / CUET prep both stand on.

13 subjects 171 chapters 177 practice quizzes NCERT syllabus

66 quiz attempts so far Dual AI-verified questions 10 quizzes free, then 10 credits

Notice Board

Students preparing for CBSE Class 11 Annual Assessment

View all

Formula of the day

Chemistry · Gaseous State · 28 formulae in the cheatsheet

Trap of the day

In v² = u² + 2as, a is negative for deceleration — keep the sign or the stopping distance comes out wrong.

Chapter 3: Trigonometric Functions — Class 11 Mathematics

Mathematics · 14 chapters

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

Chapter 3: Trigonometric Functions

Try one from this chapter

The relation between radian and degree measure is:

Summary

Trigonometric ratios of acute angles are generalised here to trigonometric functions of any angle. Angles are measured in degrees or in radians, related by \(\pi\text{ radian}=180^\circ\), and an arc of length \(l\) in a circle of radius \(r\) subtends \(\theta=l/r\) so \(l=r\theta\). Using a unit circle, \(\cos x\) and \(\sin x\) are the coordinates of a point, giving \(\sin^2 x+\cos^2 x=1\), \(1+\tan^2 x=\sec^2 x\) and \(1+\cot^2 x=\operatorname{cosec}^2 x\). Sine and cosine are periodic with period \(2\pi\); their signs vary by quadrant. The chapter derives the sum and difference formulas, e.g. \(\cos(x+y)=\cos x\cos y-\sin x\sin y\) and \(\sin(x+y)=\sin x\cos y+\cos x\sin y\), the double- and triple-angle identities such as \(\sin 2x=2\sin x\cos x\) and \(\cos 2x=1-2\sin^2 x\), and the product-to-sum and sum-to-product transformations. These identities let us evaluate functions at angles like \(15^\circ\) and prove a wide range of trigonometric equalities.

Angles: degree and radian measureArc length \(l=r\theta\)Trigonometric functions on the unit circleSigns, domain and range by quadrantSum and difference identitiesDouble, triple and half-angle formulaeProduct-to-sum and sum-to-product formulae

Key terms

Radian measure

Angle subtended at the centre by an arc equal to the radius; \(\pi\text{ rad}=180^\circ\).

Arc length relation

In a circle of radius \(r\), \(l=r\theta\) where \(\theta\) is in radians.

Pythagorean identity

\(\sin^2 x+\cos^2 x=1\), with \(1+\tan^2 x=\sec^2 x\) and \(1+\cot^2 x=\operatorname{cosec}^2 x\).

Sum formula

\(\cos(x+y)=\cos x\cos y-\sin x\sin y\); \(\sin(x+y)=\sin x\cos y+\cos x\sin y\).

Double angle

\(\sin 2x=2\sin x\cos x\); \(\cos 2x=\cos^2 x-\sin^2 x=1-2\sin^2 x\).

Periodicity

\(\sin\) and \(\cos\) repeat every \(2\pi\); \(\tan\) and \(\cot\) repeat every \(\pi\).

Important questions

Explore interactively

Key-term flashcards

Flip cards · mark known · keyboard ← → and Space

6 cards

Term1 / 6

Radian measure

Tap to reveal

Meaning1 / 6

Angle subtended at the centre by an arc equal to the radius; \(\pi\text{ rad}=180^\circ\).

Tap to flip back

Tap card to flip

Keyboard: ← → to move · Space to flip

Practice quiz · Trigonometric Functions

Score on this chapter, climb the leaderboard, and get an AI diagnosis of your mistakes.

Dual AI-verified questions Real exam pattern 2 quizzes free, then 10 credits per quiz

Trigonometric Functions

Maths 10 Qs · ~10 min