# Definitions, simple applications, and graphs of trigonometric  (Page 2/7)

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$\begin{array}{ccc}\hfill \angle A& =& \angle D\hfill \\ \hfill \angle B& =& \angle E\hfill \\ \hfill \angle C& =& \angle F\hfill \end{array}$

## Investigation : ratios of similar triangles

In your exercise book, draw three similar triangles of different sizes, but each with $\stackrel{^}{A}={30}^{\circ }$ ; $\stackrel{^}{B}={90}^{\circ }$ and $\stackrel{^}{C}={60}^{\circ }$ . Measure angles and lengths very accurately in order to fill in the table below (round answers to one decimal place).

 Dividing lengths of sides (Ratios) $\frac{AB}{BC}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{AB}{AC}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{CB}{AC}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{A}^{\text{'}}{B}^{\text{'}}}{{B}^{\text{'}}{C}^{\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{A}^{\text{'}}{B}^{\text{'}}}{{A}^{\text{'}}{C}^{\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{C}^{\text{'}}{B}^{\text{'}}}{{A}^{\text{'}}{C}^{\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{A}^{\text{'}\text{'}}{B}^{\text{'}\text{'}}}{{B}^{\text{'}\text{'}}{C}^{\text{'}\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{A}^{\text{'}\text{'}}{B}^{\text{'}\text{'}}}{{A}^{\text{'}\text{'}}{C}^{\text{'}\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$ $\frac{{C}^{\text{'}\text{'}}{B}^{\text{'}\text{'}}}{{A}^{\text{'}\text{'}}{C}^{\text{'}\text{'}}}=\phantom{\rule{42.67912pt}{0ex}}$

What observations can you make about the ratios of the sides?

These equal ratios are used to define the trigonometric functions.

Note: In algebra, we often use the letter $x$ for our unknown variable (although we can use any other letter too, such as $a$ , $b$ , $k$ , etc). In trigonometry, we often use the Greek symbol $\theta$ for an unknown angle (we also use $\alpha$ , $\beta$ , $\gamma$ etc).

## Definition of the trigonometric functions

We are familiar with a function of the form $f\left(x\right)$ where $f$ is the function and $x$ is the argument. Examples are:

$\begin{array}{ccc}\hfill f\left(x\right)& =& {2}^{x}\phantom{\rule{2.em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{(exponential}\phantom{\rule{4.pt}{0ex}}\text{function)}\hfill \\ \hfill g\left(x\right)& =& x+2\phantom{\rule{2.em}{0ex}}\text{(linear}\phantom{\rule{4.pt}{0ex}}\text{function)}\hfill \\ \hfill h\left(x\right)& =& 2{x}^{2}\phantom{\rule{2.em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{(parabolic}\phantom{\rule{4.pt}{0ex}}\text{function)}\hfill \end{array}$

The basis of trigonometry are the trigonometric functions . There are three basic trigonometric functions:

1. sine
2. cosine
3. tangent

These are abbreviated to:

1. sin
2. cos
3. tan

These functions are defined from a right-angled triangle , a triangle where one internal angle is 90 ${}^{\circ }$ .

Consider a right-angled triangle.

In the right-angled triangle, we refer to the lengths of the three sides according to how they are placed in relation to the angle $\theta$ . The side opposite to the right angle is labelled the hypotenus , the side opposite $\theta$ is labelled opposite , the side next to $\theta$ is labelled adjacent . Note that the choice of non-90 degree internal angle is arbitrary. You can choose either internal angle and then define the adjacent and opposite sides accordingly. However, the hypotenuse remains the same regardless of which internal angle you are referring to.

We define the trigonometric functions, also known as trigonometric identities, as:

$\begin{array}{ccc}\hfill sin\theta & =& \frac{opposite}{hypotenuse}\hfill \\ \hfill cos\theta & =& \frac{adjacent}{hypotenuse}\hfill \\ \hfill tan\theta & =& \frac{opposite}{adjacent}\hfill \end{array}$

These functions relate the lengths of the sides of a right-angled triangle to its interior angles.

One way of remembering the definitions is to use the following mnemonic that is perhaps easier to remember:

 S illy O ld H ens $\mathbf{S}\mathrm{in}=\frac{\mathbf{O}\mathrm{pposite}}{\mathbf{H}\mathrm{ypotenuse}}$ C ackle A nd H owl $\mathbf{C}\mathrm{os}=\frac{\mathbf{A}\mathrm{djacent}}{\mathbf{H}\mathrm{ypotenuse}}$ T ill O ld A ge $\mathbf{T}\mathrm{an}=\frac{\mathbf{O}\mathrm{pposite}}{\mathbf{A}\mathrm{djacent}}$

You may also hear people saying Soh Cah Toa. This is just another way to remember the trig functions.

The definitions of opposite, adjacent and hypotenuse are only applicable when you are working with right-angled triangles! Always check to make sure your triangle has a right-angle before you use them, otherwise you will get the wrong answer. We will find ways of using our knowledge of right-angled triangles to deal with the trigonometry of non right-angled triangles in Grade 11.

## Investigation : definitions of trigonometric functions

1. In each of the following triangles, state whether $a$ , $b$ and $c$ are the hypotenuse, opposite or adjacent sides of the triangle with respect to the marked angle.
2. Complete each of the following, the first has been done for you
$\begin{array}{ccc}\hfill a\right)\phantom{\rule{1.em}{0ex}}sin\stackrel{^}{A}& =& \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{CB}{AC}\hfill \\ \hfill b\right)\phantom{\rule{1.em}{0ex}}cos\stackrel{^}{A}& =& \\ \hfill c\right)\phantom{\rule{1.em}{0ex}}tan\stackrel{^}{A}& =\end{array}$
$\begin{array}{ccc}& d\right)& \phantom{\rule{1.em}{0ex}}sin\stackrel{^}{C}=\hfill \\ & e\right)& \phantom{\rule{1.em}{0ex}}cos\stackrel{^}{C}=\hfill \\ & f\right)& \phantom{\rule{1.em}{0ex}}tan\stackrel{^}{C}=\hfill \end{array}$
3. Complete each of the following without a calculator:
$\begin{array}{ccc}\hfill sin60& =& \\ \hfill cos30& =& \\ \hfill tan60& =\end{array}$
$\begin{array}{ccc}\hfill sin45& =& \\ \hfill cos45& =& \\ \hfill tan45& =\end{array}$

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