7.1 The trig functions and 2-d problems

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 can be 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 labeled the hypotenuse , the side opposite $\theta$ is labeled opposite , the side next to $\theta$ is labeled 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 (because it is ALWAYS opposite the right angle and ALWAYS the longest side).

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' theorem. The square of the hypotenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.