8.1 The trig functions

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:

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:

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:

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

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

   

   $$\begin{array}{ccc}\hfill a)sin\widehat{A}& =& \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{CB}{AC}\hfill \\ \hfill b)cos\widehat{A}& =& \\ \hfill c)tan\widehat{A}& =\end{array}$$
   $$\begin{array}{ccc}& d)& sin\widehat{C}=\hfill \\ & e)& cos\widehat{C}=\hfill \\ & f)& tan\widehat{C}=\hfill \end{array}$$
3. Complete each of the following without a calculator:

   

   $$\begin{array}{ccc}\hfill sin60& =& \\ \hfill cos30& =& \\ \hfill tan60& =\end{array}$$

   

   $$\begin{array}{ccc}\hfill sin45& =& \\ \hfill cos45& =& \\ \hfill tan45& =\end{array}$$

For most angles $\theta$ , it is very difficult to calculate the values of $sin\theta$ , $cos\theta$ and $tan\theta$ . One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorum. The square of the hypothenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.

These values are useful when asked to solve a problem involving trig functions without using a calculator.

Find the length of x in the following triangle.

1. Identify the trig identity that you need

   In this case you have an angle ( ${50}^{\circ }$ ), the opposite side and the hypotenuse.

   So you should use $sin$

   $$sin{50}^{\circ }=\frac{x}{100}$$
2. Rearrange the question to solve for $x$
   $$\Rightarrow x=100\times sin{50}^{\circ }$$
3. Use your calculator to find the answer

   Use the sin button on your calculator

   $$\Rightarrow x=76.6\mathrm{m}$$

Find the value of $\theta$ in the following triangle.

1. Identify the trig identity that you need

   In this case you have the opposite side and the hypotenuse to the angle $\theta$ .

   So you should use $tan$

   $$tan\theta =\frac{50}{100}$$
2. Calculate the fraction as a decimal number
   $$\Rightarrow tan\theta =0.5$$
3. Use your calculator to find the angle

   Since you are finding the angle ,

   use ${tan}^{-1}$ on your calculator

   Don't forget to set your calculator to `deg' mode!

   $$\Rightarrow \theta =26.6^{\circ }$$

The following videos provide a summary of what you have learnt so far.

Finding lengths

Find the length of the sides marked with letters. Give answers correct to 2 decimal places.