Euler's formula and Trigonometric identities
24 Feb 2018 at 1:53:24 pm

## Trigonometric Identities for Sine & Cosine

Most of the trigonometric identities for the sine and cosine functions can be derived algebraically by using Euler's formula.

There is no need to memorise them! Learning to derive them is much easier, and it sticks in memory for much longer.

The key to this is Euler's formula, according to which the sine and cosine functions are the real and imaginary parts respectively of the complex exponential function.

## Complex Exponential Function

Imagine a particle orbiting a point counterclockwise in a circular path and at a constant angular speed.

Let ${\Omega }$ denote the angular speed of the particle and ${R}$ the radius of the circular path.

Then, the particle's position can be described with the complex number:

${R}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}+{j}\phantom{\rule{0.2em}{0ex}}{R}\phantom{\rule{0.2em}{0ex}}sin{\Omega }{t}$

where the variable ${t}$ denotes the time.

The real part of this complex number denotes the x-coordinate of the particle, the imaginary part denotes the y-coordinate, which are respectively the projections of the particle's position on to the x and y axes of a rectangular coordinate system situated at the centre of the circle.

From Euler's formula, see below, we can see that the position of the particle can be described compactly with the complex exponential function.

${R}\phantom{\rule{0.2em}{0ex}}{{e}}^{{j}{\Omega }{t}}$

The constant ${\Omega }$ denotes the angular speed and the variable ${t}$ denotes the time.

If we replace ${\Omega }$ with $-{\Omega }$, then we get the complex exponential function describing the position of a clockwise-orbiting particle.

${R}\phantom{\rule{0.2em}{0ex}}{{e}}^{-{j}{\Omega }{t}}$

## Euler's Formula

Euler's formula states that $cos{\Omega }{t}$ and $sin{\Omega }{t}$ functions are the real and imaginary parts respectively of the complex exponential function ${{e}}^{{j}{\Omega }{t}}$:

(1.1)   ${{e}}^{{j}{\Omega }{t}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}+{j}\phantom{\rule{0.1em}{0ex}}sin{\Omega }{t}$

We can now derive a second formula by replacing ${\Omega }$ with $-{\Omega }$ in (1.1) and by remembering that the cosine function is an even function and the sine function is an odd function:

(1.2)   ${{e}}^{-{j}{\Omega }{t}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}-{j}\phantom{\rule{0.1em}{0ex}}sin{\Omega }{t}$

By using (1.1) and (1.2) above, we can algebraically compute $cos{\Omega }{t}$ and $sin{\Omega }{t}$, the real and imaginary parts, respectively, of the complex exponential function.

(2.1)   $cos{\Omega }{t}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{\Omega }{t}}+{{e}}^{-{j}{\Omega }{t}}}{{2}}$

(2.2)   $sin{\Omega }{t}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{\Omega }{t}}-{{e}}^{-{j}{\Omega }{t}}}{{2}{j}}$

Formulas (2.1) and (2.2) are quite useful. We can employ them to derive other formulas and trigonometric identities involving the sine and cosine functions. For example, we can easily find the derivates of the sine and cosine functions. This is shown in the next two sections.

## Derivative of the cosine function

Start by taking the derivative of (2.1)

$\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}=\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{\Omega }{t}}+{{e}}^{-{j}{\Omega }{t}}}{{2}}$

$\phantom{\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}}={j}{\Omega }\frac{{{e}}^{{j}{\Omega }{t}}-{{e}}^{-{j}{\Omega }{t}}}{{2}}$

$\phantom{\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}}={\Omega }\frac{-{{e}}^{{j}{\Omega }{t}}+{{e}}^{-{j}{\Omega }{t}}}{{2}{j}}$

$\phantom{\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}}=-{\Omega }sin{\Omega }{t}$

$\therefore \phantom{\rule{0.2em}{0ex}}\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}cos{\Omega }{t}=-{\Omega }sin{\Omega }{t}$

## Derivative of the sine function

Start by taking the derivative of (2.2)

$\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}sin{\Omega }{t}=\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{\Omega }{t}}-{{e}}^{-{j}{\Omega }{t}}}{{2}{j}}$

$\phantom{\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}sin{\Omega }{t}}={j}{\Omega }\frac{{{e}}^{{j}{\Omega }{t}}+{{e}}^{-{j}{\Omega }{t}}}{{2}{j}}$

$\phantom{\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}sin{\Omega }{t}}={\Omega }cos{\Omega }{t}$

$\therefore \phantom{\rule{0.2em}{0ex}}\frac{{d}}{{d}{t}}\phantom{\rule{0.2em}{0ex}}sin{\Omega }{t}={\Omega }cos{\Omega }{t}$

## Core Trigonometric Identities

For a moment, recall the identities (1.1), (1.2), (2.1), and (2.2) given above.

Because we don't need the time anymore, we can hide the time variable ${t}$ in them. If we substitute ${A}$ for ${\Omega }{t}$ in them, we get the following core identities. They look simpler because the time variable ${t}$ is hidden away.

(3.1)   ${{e}}^{{j}{A}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}cos{A}+{j}\phantom{\rule{0.1em}{0ex}}sin{A}$

(3.2)   ${{e}}^{-{j}{A}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}cos{A}-{j}\phantom{\rule{0.1em}{0ex}}sin{A}$

(4.1)   $cos{A}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{A}}+{{e}}^{-{j}{A}}}{{2}}$

(4.2)   $sin{A}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{A}}-{{e}}^{-{j}{A}}}{{2}{j}}$

(5.1)   ${{e}}^{{j}{\left(}{A}+{B}{\right)}}={{e}}^{{j}{A}}{{e}}^{{j}{B}}$

(5.2)   ${{e}}^{{j}{\left(}{A}-{B}{\right)}}={{e}}^{{j}{A}}{{e}}^{-{j}{B}}$

By using these core identities, we can derive hard-to-memorize identities: for example, identities for cos(A+B), cos(A-B), sin(A+B), sin(A-B), cos2A, sin2A, etc.

For this, we first pick either (5.1) or (5.2), depending on whether we are dealing with the sum of two angles or their differences. We then rewrite the righthand side by using (3.1) and (3.2). Next, we carry out the multiplication on the righthand side. Finally, we take either the real parts or imaginary parts of both sides. We take the real parts if we are dealing with the cosine function; we take the imaginary parts if we are dealing with the sine function.

## Sum and Difference Identities

Identities for the sum of two angles and difference of two angles can be computed from (5.1) and (5.2).

For example, the identity for sin(A-B) can be computed as follows.

Since sin(A-B) is the imaginary part of (5.2), we rewrite the righthand side of (5.2) by using the identities (3.1) and (3.2).

${{e}}^{{j}{\left(}{A}-{B}{\right)}}={{e}}^{{j}{A}}{{e}}^{-{j}{B}}$

${{e}}^{{j}{\left(}{A}-{B}{\right)}}={\left(}cos{A}+{j}sin{A}{\right)}{\left(}cos{B}-{j}sin{B}{\right)}$

We then perform the multiplication and take the imaginary parts of both sides to get the desired identity for sin(A-B).

$\therefore sin\phantom{\rule{0.2em}{0ex}}{\left(}{A}-{B}{\right)}=sin{A}cos{B}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}cos{A}sin{B}$

## Double-Angle Identities

Double-Angle identities also can be computed from (5.1) and (5.2).

For example, we compute the identity for cos(2A) as follows.

Because cos(2A) equals cos(A+B) with B=A, we first produce an intermediate result for cos(A+B). For this, we rewrite the righthand side of (5.1) by using the identity (3.1).

${{e}}^{{j}{\left(}{A}+{B}{\right)}}={{e}}^{{j}{A}}{{e}}^{{j}{B}}$

${{e}}^{{j}{\left(}{A}+{B}{\right)}}={\left(}cos{A}+{j}sin{A}{\right)}{\left(}cos{B}+{j}sin{B}{\right)}$

We then perform the multiplication and take the real parts of both sides to get the intermediate result.

$\therefore \phantom{\rule{0.2em}{0ex}}cos\phantom{\rule{0.2em}{0ex}}{\left(}{A}+{B}{\right)}=cos{A}cos{B}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}sin{A}sin{B}$

Finally, we set B=A in the intermediate result to get the desired identity for cos(2A).

$\therefore \phantom{\rule{0.2em}{0ex}}cos\phantom{\rule{0.2em}{0ex}}{\left(}{2}{A}{\right)}={cos}^{{2}}{A}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{sin}^{{2}}{A}$

## Product Identities

Product identities can be computed by multiplying (4.1) and (4.2).

For example, the product of cosA and sinA can be computed as follows.

$cos{A}\phantom{\rule{0.2em}{0ex}}sin{A}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{A}}+{{e}}^{-{j}{A}}}{{2}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{A}}-{{e}}^{-{j}{A}}}{{2}{j}}$

$\phantom{cos{A}\phantom{\rule{0.2em}{0ex}}sin{A}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{{e}}^{{j}{2}{A}}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{{e}}^{-{j}{2}{A}}}{{4}{j}}$

$\phantom{cos{A}\phantom{\rule{0.2em}{0ex}}sin{A}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{{1}}{{2}}\phantom{\rule{0.2em}{0ex}}sin\phantom{\rule{0.2em}{0ex}}{2}{A}$

## Note

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