Introduction to Trigonometry · Class 10 Mathematics: quick revision, 49 bites

Why ratios work

Size does not matter

Two right triangles with the same acute angle are similar, so their side ratios are identical.

Takeaway. That is why the ratios are worth naming.

Naming from the angle

Which side is which

Takeaway. Opposite and adjacent SWAP if you switch angles.

The three main ratios

SOH-CAH-TOA

Takeaway. Three letters, three ratios.

The three reciprocals

Just upside down

Takeaway. Six ratios, but only three ideas.

sec goes with cos

The pairing is crossed

coSecant is the reciprocal of Sine; seCant is the reciprocal of Cosine.

Takeaway. Match the THIRD letters.

A useful link

Tan from the other two

Takeaway. And $\cot A = \frac{\cos A}{\sin A}$.

The size of sin and cos

Never more than one

The hypotenuse is the longest side, so the fraction cannot exceed 1.

Takeaway. An answer of $\frac54$ is a signal to check your working.

The size of sec and cosec

Never less than one

Turning a number at most 1 upside down gives a number at least 1.

Takeaway. tan and cot have no such limit.

What sin²A means

Square the ratio, not the angle

$\sin^2 A$ means $(\sin A)^2$.

Takeaway. The power sits on the ratio.

What sin⁻¹A means

Not a reciprocal

$\sin^{-1} A$ is the INVERSE FUNCTION — 'which angle has this sine?'

Takeaway. That is why cosecant has its own name.

The standard method

One ratio gives all six

Takeaway. Three steps, every time.

Given sin A = 3/5

A worked case

Takeaway. Pythagoras supplies the missing side.

Given tan A = 1/2

Surds are fine

Opposite 1, adjacent 2, so hypotenuse $= \sqrt5$.

Takeaway. No need to rationalise unless asked.

Similarity again

Why you may pick the simplest sides

$\tan A = \frac12$ could come from sides 1 and 2, or 50 and 100.

Takeaway. The whole topic rests on this.

Where the marks are

Draw it, label it

A rough triangle with its three sides labelled earns most of the marks in these questions.

Takeaway. Identifying the sides IS the skill being tested.

Where 'sine' comes from

A mistranslation

Sanskrit *jya* became Arabic *jiba*, was misread as *jaib* meaning a bay, and entered Latin as *sinus*.

Takeaway. Aryabhata tabulated the values in the fifth century.

Five angles, three rows

The whole table

Takeaway. Learn this and most of the chapter follows.

Remembering the sine row

One list, not three

Write $\sqrt{\frac04}, \sqrt{\frac14}, \sqrt{\frac24}, \sqrt{\frac34}, \sqrt{\frac44}$.

Takeaway. Cosine is the same list reversed.

Why you learn only one row

Cosine is sine backwards

Reading the sine row right to left gives the cosine row.

Takeaway. And tan = sin ÷ cos, so that row is free.

Where 45° comes from

Half a square

Takeaway. Equal legs, so equal ratios.

Where 30° and 60° come from

Half an equilateral triangle

Drop a perpendicular in an equilateral triangle of side 2.

Takeaway. One triangle supplies four of the values.

The 30° and 60° values

From that triangle

Takeaway. The 30° and 60° values are each other's swap.

Where 0° and 90° come from

Flattening the triangle

No right triangle can contain a second 90° angle, so these come from imagining it collapse.

Takeaway. That is why two entries have no value.

tan 90°

Undefined, not infinity

$\tan 90°$ would divide by an adjacent side of length zero.

Takeaway. $\sec 90°$ and $\operatorname{cosec} 0°$ are undefined too.

The sin/cos swap

Which row goes up?

$\sin$ INCREASES from 0 to 1; $\cos$ DECREASES from 1 to 0.

Takeaway. Picture a nearly flat triangle: tiny angle, tiny sine.

Thirty seconds well spent

Write the table first

Copy the table into the margin before starting the question.

Takeaway. Then substitute, then simplify.

Substitute first

Evaluating an expression

Takeaway. Simplifying before substituting invites errors.

sin30 cos60 + cos30 sin60

A worked evaluation

$\frac12 \times \frac12 + \frac{\sqrt3}2 \times \frac{\sqrt3}2$

Takeaway. Exactly 1 — and that is $\sin 90°$.

2tan²45 + cos²30 − sin²60

Another one

$= 2(1)^2 + \frac34 - \frac34 = 2$.

Takeaway. Spotting equal values saves work.

sin(A + B)

Brackets are not optional

$\sin(30° + 60°) = \sin 90° = 1$, but $\sin 30° + \sin 60° \approx 1.37$.

Takeaway. Evaluate inside the bracket first.

cosec, sec and cot

The reciprocals follow

$\operatorname{cosec} 30° = 2$, $\sec 60° = 2$, $\cot 45° = 1$.

Takeaway. Except where the value is zero.

Slopes you meet

Why these angles?

A 30° ramp rises exactly half its length; a 45° roof rises as much as it runs.

Takeaway. Convenient numbers, and convenient ratios.

Identity 1

The one that matters most

Takeaway. Pythagoras divided by the hypotenuse squared.

The difference

An identity is not an equation

$\sin\theta = \frac12$ is an EQUATION — a question about which angles work.

Takeaway. You never solve an identity.

Four lines

Proving identity 1

Takeaway. The whole proof is one division.

Identity 2

Divide by cos²

Takeaway. Valid for $0° \le \theta < 90°$.

Identity 3

Divide by sin²

Takeaway. Valid for $0° < \theta \le 90°$.

The naming rule

Which division gives which

Divide by $\cos^2$ and you get SEC; divide by $\sin^2$ and you get COSEC.

Takeaway. One identity generates the other two.

Rearrangements

The forms you actually use

Takeaway. Learn the rearrangements, not just the originals.

A fatal slip

The squares are not decoration

$\sin^2\theta + \cos^2\theta = 1$ does NOT give $\sin\theta + \cos\theta = 1$.

Takeaway. Square roots do not distribute over a sum.

sec² − tan², not the reverse

Mind the order

It is $\sec^2\theta - \tan^2\theta = 1$.

Takeaway. Sec is the bigger one — it is at least 1.

The rule for proofs

One side only

Take the more complicated side and transform it until it becomes the other.

Takeaway. Write 'LHS =' and keep going.

Convert to sin and cos

The reliable rescue

Rewrite tan, cot, sec and cosec in terms of sine and cosine.

Takeaway. This move saves most stalled proofs.

tan + cot = sec cosec

A worked conversion

Takeaway. Identity 1 appeared the moment the fractions combined.

For sec − tan and friends

The conjugate trick

Takeaway. $\frac{1}{\sec - \tan} = \sec + \tan$.

Using a conjugate

Both, not one

Multiply the numerator AND the denominator by the conjugate.

Takeaway. You are multiplying by 1, in disguise.

sin⁴ − cos⁴

Factorising wins too

$= (\sin^2 + \cos^2)(\sin^2 - \cos^2)$, a difference of two squares.

Takeaway. Ordinary algebra does most of the work.

The excluded angles

Where they fail

Identity 2 fails at 90° and identity 3 at 0°, because the ratios involved are undefined there.

Takeaway. Identity 1 holds everywhere.

Why identity 1 matters

A point on a circle

$\sin^2 + \cos^2 = 1$ says that $(\cos\theta, \sin\theta)$ always lies on the unit circle.

Takeaway. The same identity runs through all later mathematics.