Trigonometry Formula List: Ratios, Table and Identities

Sin cos tan formula: the six trigonometric ratios

Take a right triangle and fix one acute angle $\theta$. The side opposite $\theta$ is the perpendicular (P), the side next to it is the base (B), and the longest side, facing the right angle, is the hypotenuse (H). The six ratios are:

So the sin theta formula is perpendicular over hypotenuse, the cos theta formula is base over hypotenuse (cos is base upon hypotenuse), and the tan theta formula is perpendicular over base. A memory line many teachers use is "Pandit Badri Prasad, Har Har Bole": the first letters pair up as P/H, B/H, P/B. The other three ratios are just the flips, so $\operatorname{cosec}\theta = \dfrac{1}{\sin\theta}$, $\sec\theta=\dfrac{1}{\cos\theta}$ and $\cot\theta=\dfrac{1}{\tan\theta}$. You will also see the useful link $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$.

How do you find the hypotenuse in trigonometry when only two legs are given? Use Pythagoras, $H^2=P^2+B^2$. If you know one side and one angle instead, rearrange a ratio: $H = \dfrac{P}{\sin\theta}$. The same right triangle idea also gives you the distance formula between two points.

Trigonometry table formula for 0, 30, 45, 60 and 90 degrees

Board papers expect you to know these values without a calculator. Here is the trigonometry table for the five standard angles.

A quick way to rebuild the sin row in the exam hall: write 0, 1, 2, 3, 4, divide each by 4 and take the square root. You get $0, \tfrac12, \tfrac{1}{\sqrt2}, \tfrac{\sqrt3}{2}, 1$. Write the same row backwards for cos. Then divide sin by cos for tan, and flip the rows for cosec, sec and cot.

Trigonometric identities

An identity is true for every angle where both sides make sense. Class 10 uses three, all coming from Pythagoras divided through by $H^2$, $B^2$ or $P^2$:

Rewrite them as you need: $\sec^2\theta-\tan^2\theta=1$ is often handy because it factors as $(\sec\theta-\tan\theta)(\sec\theta+\tan\theta)=1$. The sec 1 formula students ask about is this one, $\sec^2\theta = 1+\tan^2\theta$.

Allied angle formulas and quadrant signs

Class 11 extends the ratios to any angle using the unit circle. Angles such as $90^\circ\pm\theta$, $180^\circ\pm\theta$, $270^\circ\pm\theta$ and $360^\circ-\theta$ are called allied angles. Two rules cover all of them:

• Name rule: with 90° or 270°, sin becomes cos, tan becomes cot, sec becomes cosec (and the reverse). With 180° or 360°, the name stays.
• Sign rule: the sign is the sign of the original ratio in the quadrant where the angle lands. Only sin and cosec are positive in quadrant II, only tan and cot in quadrant III, only cos and sec in quadrant IV, and all are positive in quadrant I. Remember it as "All Silver Tea Cups".

For example, $\cos 150^\circ=\cos(180^\circ-30^\circ)=-\cos 30^\circ=-\tfrac{\sqrt3}{2}$. The $90^\circ-\theta$ row is the one you are most likely to have met before class 11; it is why $\sin 30^\circ=\cos 60^\circ$. In radians, $180^\circ=\pi$, so these rows also read $\sin(\pi-\theta)=\sin\theta$ and so on.

Compound angle formulas: sin(A+B), cos(A+B), tan(A+B)

These give the ratio of a sum or difference of two angles. They are the root of every class 11 formula that follows.

Watch the cosine line: the sign in the middle flips. The cos(a+b) formula has a minus, while $\cos(A-B)$ has a plus. The sin a+b formula keeps the same sign on both sides.

Double angle trigonometry formula: sin 2x, cos 2x and tan 2x

Put $B=A$ in the compound angle results and you get the double angle set. Searches for the sin2x formula, cos2x formula, sin2a formula, cos 2 theta formula and tan 2 theta formula all point here; only the letter changes.

The three forms of cos 2x come from swapping $\sin^2x=1-\cos^2x$ or the reverse. Pick the form that leaves only the ratio you know. In terms of tan alone:

1 - cos 2x and 1 + cos 2x

Rearranging the last two forms of cos 2x gives the power reducing pair, used all the time in integration:

Replace $2x$ by $A$ and you get the half angle version, $\cos A = 2\cos^2\tfrac{A}{2}-1$, which answers the cos a/2 search.

1 + sin x formula

Since $\sin^2\tfrac{x}{2}+\cos^2\tfrac{x}{2}=1$ and $\sin x=2\sin\tfrac{x}{2}\cos\tfrac{x}{2}$, you can write $1+\sin x$ as a perfect square:

Triple angle formulas: sin 3x, cos 3x and tan 3x

Write $3x=2x+x$, expand with the compound angle rule, and replace the 2x terms. The results are:

Notice the pattern: the sin3x formula starts with the 3 term and subtracts the cube, while the cos3x formula starts with the cube and subtracts the 3 term. A fast self check is $x=30^\circ$: $3\cdot\tfrac12-4\cdot\tfrac18=1=\sin 90^\circ$.

Product to sum and sum to product formulas

Adding and subtracting the compound angle lines turns a product of two ratios into a sum:

The last line is the sin a sin b formula. Note the order: $\cos(A-B)$ comes first. Now put $A+B=C$ and $A-B=D$ to go the other way:

Inverse trigonometry formula basics for class 12

Class 12 chapter 2, Inverse Trigonometric Functions, asks the reverse question: which angle has a given sine? To make the answer unique, each inverse function has a principal value range, for example $\sin^{-1}x\in\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$, $\cos^{-1}x\in[0,\pi]$ and $\tan^{-1}x\in\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$. The results students use most are:

Keep in mind that $\sin^{-1}x$ is not $\dfrac{1}{\sin x}$; that reciprocal is $\operatorname{cosec} x$. The derivatives of these functions are listed on the differentiation formulas page, and they turn up again as answers on the integration formulas page.

Trigonometry formula class 10 vs class 11: what is in your syllabus

If you only need the class 10 trigonometry formula list, you can stop after the identities section. Here is how the topic is split across NCERT books.

Karnataka 1st PUC students can use the class 11 list. BSc first year maths and engineering entrance papers assume all of it, plus the radian form of every result. In class 10 the heights and distances chapter mostly uses tan, since you usually know a horizontal distance and want a height.

One radian conversion worth keeping handy: $\pi\text{ rad}=180^\circ$, so $1^\circ=\tfrac{\pi}{180}$ rad. Angles in class 11 problems are often given as $\tfrac{\pi}{6}$, $\tfrac{\pi}{4}$, $\tfrac{\pi}{3}$, which are simply 30°, 45° and 60°.

Common mistakes with trig formulas

• Writing $\sin(A+B)=\sin A+\sin B$. Try A = B = 45° and you get $\sin 90^\circ=1$ on the left and $\sqrt2$ on the right, so it is wrong.
• Mixing up $\sin^2\theta$ and $\sin\theta^2$. The first squares the value of sine; the second squares the angle.
• Picking the wrong form of cos 2x. If a question gives cos x, use $2\cos^2x-1$; if it gives sin x, use $1-2\sin^2x$.
• Forgetting the sign in quadrants II to IV for allied angles. Draw a quick quadrant sketch before you write the answer.
• Using degrees in a formula that needs radians, such as arc length $l=r\theta$.
• Writing $\tan 90^\circ$ as a number. It has no value.

For the algebra steps that keep coming up inside trig proofs, like $a^2-b^2=(a-b)(a+b)$, see algebra identities. The complete list across topics sits on the maths formulas page, and the formula sheet PDF of this page's key box is available to download below.