7 Trigonometry

7.1 Sine and Cosine Rules

Figure 7.1 shows a triangle with sides a, b and c and angles subtending the sides A, B and C respectively.

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Figure 7.1: Triangle with side lengths a, b and c.

For the triangle in Figure 7.1 we can write


These equations are called the sine rule.

Note: The sine of an angle, $\theta$, is the same as the sine of the supplement of the angle, $\pi - \theta$ ($180 - \theta$). This means the sine rule fails for triangles with an obtuse angle.

As a precaution, always calculate all three angles and add the results together. If the sum is less than $\pi$ ($180$) then the angle opposite the longest side is probably obtuse.

For the same triangle we can also write

$a^2=b^2+c^2-2bc\ cos(A)$

This equation is called the cosine rule.

When angle $A=\pi/2$, so we have a right angled triangle and the cosine rule simplifies to Pythagoras's theorem.


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Figure 7.2: Right angle triangle.

For right angled triangles we can write
$sin(C)= \dfrac{c}{a}$, $cos(C)= \dfrac{b}{a}$ and $tan(C)= \dfrac{c}{b}$.

For angle $C$ side $b$ is called the adjacent, side $c$ is called the opposite and side $a$ is called the hypotenuse.

A common mnemonic for right angled triangles is SohCahToa which means
$sin(C)=\frac{opposite}{hypotenuse}$, $cos(C)=\frac{adjacent}{hypotenuse}$ and $tan(C)=\frac{opposite}{adjacent}$

A quick way to work out whether to use $sin()$ or $cos()$ to find a length is to imagine standing at angle $C$. Look along the adjacent side. We could say we were looking in the central direction and the central direction gives us $cos(C) = c/a$.

The opposite side goes sideways and sideways gives us $sin(\theta)=c/a$.

Example 7.1
Find the angles for a triangle with sides $a=4$, $b=5$ and $c=6$.
We need to use the cosine rule to find the first angle.
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$a^2$$=$$b^2+c^2-2bc\ cos(A)$
$2bc\ cos(A)$$=$$b^2+c^2-a^2$
 $$A$$$=$$cos^{-1}(\dfrac{5^2+6^2-4^2}{2 \times 5 \times 4})$
 $$A$$$=$$0.72$ radians (or $41.41^\circ$)
Similarly we get $B$
 $$B$$$=$$0.97$ radians (or $55.77^\circ$)
and we get $C$
 $$C$$$=$$1.45$ radians (or $82.82^\circ$)
Sanity Check
You might have been tempted to calculate two of the angles and subtract their sum from $\pi$ or $180^\circ$. If you did that you would have no easy way to check whether or not you had calculated three wrong angles. Better to calculate all three angles and add them together.
(or $41.41+55.77+82.82=180$)
so we can be confident our answers are correct.

7.2 Cartesian and Polar Coordinates

So far we have been sketching our diagrams on x-y axes. This system is attributed to René Descartes and is called the Cartesian coordinate system. The system allows us to describe any point on a 2D plane.

An alternative way to describe a point on a 2D plane is by giving an angle from a fixed line and a radius. This system is calle the polar coordinate system and is used a lot in navigation.

To convert from cartesian coordinates to polar coordinates we have:


$\theta = tan^{-1}(y/x)$

To convert from polar coordinates to cartesian coordinates we have:

$x=r\ cos(\theta)$

$y=r\ sin(\theta)$

7.3 Trigonometric Identities

Imagine a disc with a pen mounted near the edge. If the disc rotates about its centre the pen will draw a circle. Now, imagine there is a long strip of paper under the disc. If we pull the paper at a constant speed as the disc rotates, instead of a circle the pen will draw a wiggley line. The line is a sinusoidal curve and this is why sines, cosines and tangents are called circular functions.

Pythagoras's theorem states $x^2+y^2=r^2$. If we divide both sides by $r^2$ we get

Looking at figure 7.3 we can see $sin(\theta)= \dfrac{y}{r}$ and $cos(\theta)= \dfrac{x}{r}$ substituting these into our equation we get:


The angle $\theta$ was not used in our derivation which means $cos^2(\theta)+sin^2(\theta)=1$ is true for any value of $\theta$. This means the expression is more than an equation, it is an identity.

Here are some other trigonometric identities

$ \cos(\theta + \phi) = \cos(\theta)\cos(\phi)- \sin(\theta)\sin(\phi)$

$ \cos(\theta - \phi) = \cos(\theta)\cos(\phi)+ \sin(\theta)\sin(\phi)$

$ \sin(\theta + \phi) = \sin(\theta)\cos(\phi)+ \cos(\theta)\sin(\phi)$

$ \sin(\theta - \phi) = \sin(\theta)\cos(\phi)- \cos(\theta)\sin(\phi)$

$ \tan(\theta + \phi) = \dfrac{\tan(\theta)+ \tan(\phi)}{1-\tan(\theta) \tan(\phi)} $

$ \tan(\theta - \phi) = \dfrac{\tan(\theta)- \tan(\phi)}{1+\tan(\theta) \tan(\phi)} $

If we let $\theta = \phi$ we get

$ \cos(2\theta) = \cos^2(\theta)- \sin^2(\theta)$

$ \sin(2\theta) = 2 \sin(\theta)\cos(\theta)$

$ \tan(2\theta) = \dfrac{2\tan \theta}{1-\tan^2 \theta} $