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.

Trigonometric Functions

Class 11 Mathematics · Chapter 3

163 members in the CBSE community

View all

CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

View all 44 amazing Marks Booster kits →

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 \(\pi\).

Extra questions & answers

Try one from this chapter

The relation between radian and degree measure is:

Flashcards & interactive practice