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Overview

Trigonometric identities are some of the most useful tools in olympiad trigonometry. They allow us to transform complicated expressions into simpler ones.

These techniques are especially useful when expressions contain products of sines and cosines, repeated angles, or long sums involving trigonometric terms.

The main goal is not to memorise identities blindly, but to understand when and why each identity is useful.

Motivation

Consider the expression

sin⁡20∘cos⁡40∘.\sin 20^\circ \cos 40^\circ.

At first, this product does not look easy to simplify. However, product to sum identities allow us to rewrite it as a sum:

sin⁡Acos⁡B=12(sin⁡(A+B)+sin⁡(A−B)).\sin A \cos B = \frac{1}{2}(\sin(A+B)+\sin(A-B)).

Using A=20∘A=20^\circ and B=40∘B=40^\circ, we get

sin⁡20∘cos⁡40∘=12(sin⁡60∘+sin⁡(−20∘)).\sin 20^\circ \cos 40^\circ = \frac{1}{2}(\sin 60^\circ+\sin(-20^\circ)).

Since sin⁡(−20∘)=−sin⁡20∘\sin(-20^\circ)=-\sin 20^\circ, this becomes

12(sin⁡60∘−sin⁡20∘).\frac{1}{2}(\sin 60^\circ-\sin 20^\circ).

A product has been changed into a sum, which is often easier to combine with other terms.

This is the central idea behind many trigonometric identities, rewrite an expression into a form where cancellation, symmetry, or known values become visible.

Angle Addition Formulas

Most of the identities in this module come from the angle addition formulas.

For all angles AA and BB,

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B

and

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B.\sin(A-B)=\sin A\cos B-\cos A\sin B.

Also,

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)=\cos A\cos B-\sin A\sin B

and

cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B.\cos(A-B)=\cos A\cos B+\sin A\sin B.

These four formulas are the foundation for product to sum, sum to product, double angle, and triple angle identities.

Product to Sum Identities

Product to sum identities rewrite products of trigonometric functions as sums.

They are especially useful when several product terms appear together.

Main Identities

sin⁡Acos⁡B=12(sin⁡(A+B)+sin⁡(A−B))\sin A\cos B = \frac{1}{2}(\sin(A+B)+\sin(A-B))
cos⁡Asin⁡B=12(sin⁡(A+B)−sin⁡(A−B))\cos A\sin B = \frac{1}{2}(\sin(A+B)-\sin(A-B))
cos⁡Acos⁡B=12(cos⁡(A+B)+cos⁡(A−B))\cos A\cos B = \frac{1}{2}(\cos(A+B)+\cos(A-B))
sin⁡Asin⁡B=12(cos⁡(A−B)−cos⁡(A+B)).\sin A\sin B = \frac{1}{2}(\cos(A-B)-\cos(A+B)).

These identities come directly from adding or subtracting the angle addition formulas.

Deriving a Product to Sum Identity

Let us derive one example.

From the sine addition formulas,

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B

and

sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B.\sin(A-B)=\sin A\cos B-\cos A\sin B.

Adding these equations gives

sin⁡(A+B)+sin⁡(A−B)=2sin⁡Acos⁡B.\sin(A+B)+\sin(A-B)=2\sin A\cos B.

Therefore,

sin⁡Acos⁡B=12(sin⁡(A+B)+sin⁡(A−B)).\sin A\cos B = \frac{1}{2}(\sin(A+B)+\sin(A-B)).

The other product to sum identities are derived similarly.

Sum to Product Identities

The same identities can be read in reverse. This gives sum to product identities.

These are useful when a problem contains sums such as sin⁡A+sin⁡B\sin A+\sin B or cos⁡A−cos⁡B\cos A-\cos B.

Main Identities

sin⁡X+sin⁡Y=2sin⁡(X+Y2)cos⁡(X−Y2)\sin X+\sin Y = 2\sin\left(\frac{X+Y}{2}\right) \cos\left(\frac{X-Y}{2}\right)
sin⁡X−sin⁡Y=2cos⁡(X+Y2)sin⁡(X−Y2)\sin X-\sin Y = 2\cos\left(\frac{X+Y}{2}\right) \sin\left(\frac{X-Y}{2}\right)
cos⁡X+cos⁡Y=2cos⁡(X+Y2)cos⁡(X−Y2)\cos X+\cos Y = 2\cos\left(\frac{X+Y}{2}\right) \cos\left(\frac{X-Y}{2}\right)
cos⁡X−cos⁡Y=−2sin⁡(X+Y2)sin⁡(X−Y2).\cos X-\cos Y = -2\sin\left(\frac{X+Y}{2}\right) \sin\left(\frac{X-Y}{2}\right).

These often appear in problems involving long sums or equations where factorisation is needed.

Double angle Identities

Double angle identities express trigonometric functions of 2x2x in terms of functions of xx.

Sine Double angle Formula

Using

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B,\sin(A+B)=\sin A\cos B+\cos A\sin B,

and setting A=B=xA=B=x, we get

sin⁡2x=2sin⁡xcos⁡x.\sin 2x=2\sin x\cos x.

Cosine Double angle Formula

Using

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B,\cos(A+B)=\cos A\cos B-\sin A\sin B,

and setting A=B=xA=B=x, we get

cos⁡2x=cos⁡2x−sin⁡2x.\cos 2x=\cos^2 x-\sin^2 x.

Using

sin⁡2x+cos⁡2x=1,\sin^2 x+\cos^2 x=1,

we also get the equivalent forms

cos⁡2x=2cos⁡2x−1\cos 2x=2\cos^2 x-1

and

cos⁡2x=1−2sin⁡2x.\cos 2x=1-2\sin^2 x.

Tangent Double angle Formula

Since

tan⁡2x=sin⁡2xcos⁡2x,\tan 2x=\frac{\sin 2x}{\cos 2x},

we get

tan⁡2x=2tan⁡x1−tan⁡2x,\tan 2x = \frac{2\tan x}{1-\tan^2 x},

provided the expressions are defined.

Auxiliary Angle Method

The auxiliary angle method is used to rewrite expressions of the form

asin⁡x+bcos⁡xa\sin x+b\cos x

as a single sine or cosine expression.

The key idea is to write

asin⁡x+bcos⁡x=Rsin⁡(x+ϕ).a\sin x+b\cos x=R\sin(x+\phi).

Using the angle addition formula,

Rsin⁡(x+ϕ)=Rsin⁡xcos⁡ϕ+Rcos⁡xsin⁡ϕ.R\sin(x+\phi) = R\sin x\cos\phi+R\cos x\sin\phi.

Comparing coefficients with

asin⁡x+bcos⁡x,a\sin x+b\cos x,

we need

Rcos⁡ϕ=aR\cos\phi=a

and

Rsin⁡ϕ=b.R\sin\phi=b.

Squaring and adding gives

R2=a2+b2,R^2=a^2+b^2,

so

R=a2+b2.R=\sqrt{a^2+b^2}.

Also,

tan⁡ϕ=ba,\tan\phi=\frac{b}{a},

provided a≠0a\neq 0.

Therefore,

asin⁡x+bcos⁡x=Rsin⁡(x+ϕ),a\sin x+b\cos x = R\sin(x+\phi),

where

R=a2+b2.R=\sqrt{a^2+b^2}.

Since the maximum value of sin⁡(x+ϕ)\sin(x+\phi) is 11 and the minimum value is −1-1, the maximum value of

asin⁡x+bcos⁡xa\sin x+b\cos x

is

RR

and the minimum value is

−R.-R.

Example

Find the maximum and minimum values of

3sin⁡x+4cos⁡x.3\sin x+4\cos x.

Here,

a=3a=3

and

b=4.b=4.

As such

R=32+42=5.R=\sqrt{3^2+4^2}=5.

So

3sin⁡x+4cos⁡x3\sin x+4\cos x

has maximum value 55 and minimum value −5-5.

We can also write it as

3sin⁡x+4cos⁡x=5sin⁡(x+ϕ),3\sin x+4\cos x=5\sin(x+\phi),

where

cos⁡ϕ=35\cos\phi=\frac{3}{5}

and

sin⁡ϕ=45.\sin\phi=\frac{4}{5}.

Part 2:

See continuation in Part 2!

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