Trigonometry

A ladder leans against a wall. Knowing its length and the angle it makes with the ground, we can find how high it reaches — without measuring. Trigonometry does exactly this: it links the angles of a right triangle to the ratios of its sides. The six ratios — $sin, cos, tan, cot, sec, cosec$ — and the relations between them are the foundation of the whole chapter.

sin θ = \frac{लंब}{कर्ण}, cos θ = \frac{आधार}{कर्ण}, tan θ = \frac{लंब}{आधार}; sin^{2} θ + cos^{2} θ = 1

The basic ratios and the basic identity

1

Measuring angles

Degrees, minutes and radians

The degree–minute–second system, radian measure and converting between them.

Degrees and radians

π

radians

= 18 0^{\circ}

The conversion rules and the arc–angle relation.

Sexagesimal system: $1^{\circ} = 6 0^{'}$ (minutes), $1^{'} = 6 0^{''}$ (seconds).
Radian: the angle made at the centre by an arc of length 1 in a circle of radius 1. $2 π$ radians $= 36 0^{\circ}$ , so $π$ radians $= 18 0^{\circ}$ and $1$ radian $\approx 5 7^{\circ} 1 6^{'} 2 2^{''}$ .
Degrees → radians: $\times \frac{π}{180}$ ; radians → degrees: $\times \frac{180}{π}$ (or put $π = 18 0^{\circ}$ ).
Arc and angle: $θ (radians) = \frac{arc s}{radius r}$ , that is $s = r θ$ .

Example 1 · Basic

Convert 105° into radians.

Solution, step by step

1 $105 \times \frac{π}{180} = \frac{105 π}{180}$ .
2Divide by 15: $\frac{7 π}{12}$ radians

Answer $\frac{7 π}{12}$

Example 2 · Standard

What angle (in degrees) does an arc 11 cm long subtend at the centre of a circle of radius 7 cm? (

π = \frac{22}{7}

)

Solution, step by step

1 $θ = \frac{11}{7}$ radians.
2In degrees: $\frac{11}{7} \times \frac{180 \times 7}{22} = 9 0^{\circ}$

Answer 90°

BasicTry it yourself first

Convert 36° into radians.

BasicTry it yourself first

Convert 45° into radians.

StandardTry it yourself first

Convert 72° into radians.

StandardTry it yourself first

Convert 150° into radians.

StandardTry it yourself first

Convert 210° into radians.

AdvancedTry it yourself first

Convert 225° into radians.

AdvancedTry it yourself first

Convert 18° into radians.

BasicTry it yourself first

Convert

\frac{π}{2}

radians into degrees.

StandardTry it yourself first

Convert

\frac{3 π}{4}

radians into degrees.

StandardTry it yourself first

Convert

\frac{2 π}{3}

radians into degrees.

StandardTry it yourself first

Convert

\frac{5 π}{6}

radians into degrees.

AdvancedTry it yourself first

Convert

\frac{7 π}{9}

radians into degrees.

AdvancedTry it yourself first

Convert

\frac{11 π}{12}

radians into degrees.

BasicTry it yourself first

Convert 36° 30′ into decimal degrees.

StandardTry it yourself first

Convert 25° 15′ into decimal degrees.

StandardTry it yourself first

Convert 42° 45′ into decimal degrees.

AdvancedTry it yourself first

Convert 18° 24′ into decimal degrees.

StandardTry it yourself first

Write 25.5° in degrees and minutes.

StandardTry it yourself first

Write 12.25° in degrees and minutes.

AdvancedTry it yourself first

Write 40.75° in degrees and minutes.

AdvancedTry it yourself first

Write 33.2° in degrees and minutes.

StandardTry it yourself first

What angle in degrees does an arc 22 cm long subtend at the centre of a circle of radius 14 cm? (

π = \frac{22}{7}

)

StandardTry it yourself first

What angle in degrees does an arc 11 cm long subtend at the centre of a circle of radius 14 cm? (

π = \frac{22}{7}

)

StandardTry it yourself first

What angle in degrees does an arc 44 cm long subtend at the centre of a circle of radius 21 cm? (

π = \frac{22}{7}

)

AdvancedTry it yourself first

What angle in degrees does an arc 33 cm long subtend at the centre of a circle of radius 14 cm? (

π = \frac{22}{7}

)

AdvancedTry it yourself first

What angle in degrees does an arc 66 cm long subtend at the centre of a circle of radius 21 cm? (

π = \frac{22}{7}

)

AdvancedTry it yourself first

What angle in degrees does an arc 22 cm long subtend at the centre of a circle of radius 21 cm? (

π = \frac{22}{7}

)

AdvancedTry it yourself first

What angle in degrees does an arc 77 cm long subtend at the centre of a circle of radius 42 cm? (

π = \frac{22}{7}

)

2

Ratios and identities

Six ratios, three identities

The six ratios of a right triangle, their reciprocal relations and the three basic identities.

The six ratios

Opposite, adjacent, hypotenuse

The names of the sides relative to $θ$ and the ratios; getting the rest from one ratio.

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Identities

sin^{2} + cos^{2} = 1

and its companions

The three basic identities and the questions built on them.

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3

Values at standard angles

0^{\circ}, 3 0^{\circ}, 4 5^{\circ}, 6 0^{\circ}, 9 0^{\circ}

The table, signs by quadrant, reducing large and negative angles to a reference angle, complementary angles.

The table of standard angles

Learn it once, use it always

$sin, cos, tan$ at five angles and the expressions built from them.

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Large angles, negative angles and signs

Reference angle + the quadrant’s sign

Bringing any angle to a reference angle in $0^{\circ}$ – $9 0^{\circ}$ to find its value.

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4

Sum–difference and double-angle formulas

Formulas for combining angles

$sin (A \pm B)$ , $cos (A \pm B)$ , $tan (A \pm B)$ , the double-angle formulas and the values at $1 5^{\circ}$ and $7 5^{\circ}$ .

The formulas and their use

Spot the formula, then use the table

Sum–difference, double-angle and product-to-sum formulas.

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5

Maxima, minima and special results

Limits and chain products

The limits of $sin, cos$ , the maximum and minimum of $a sin θ + b cos θ$ and the results for chain products.

Limits and chain products

- 1 \leq sin θ \leq 1

The range of values and special results.

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6

Traps and fast tricks

Slips · tricks

Where marks go in trigonometry.

The commonest mistakes

Seven traps

The slips that cost marks.

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Fast tricks

Recognise and apply

Habits for speed.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Measuring angles

Degrees and radians

Ratios and identities

The six ratios

Identities

Values at standard angles

The table of standard angles

Large angles, negative angles and signs

Sum–difference and double-angle formulas

The formulas and their use

Maxima, minima and special results

Limits and chain products

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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