Trigonometry/Cosine and Sine

Two Approaches

The cosine and sine functions relate the angles in right triangles to the ratios of lengths of sides.

There are two usual approaches to introducing the cosine and sine functions.

• In one approach the sine and cosine function are defined in terms of the right angle triangles. This works fine for angles between 0^o and 90^o. Later on the definition has to be extended to angles outside that range.
• An alternative approach introduces sine and cosine in terms of 'the unit circle'. This approach is a little more sophisticated but has the advantage that it works for all angles.

The two approaches amount to exactly the same thing in the end. We prefer to deal with the full range of angles from the start, which is why in the previous exercise we had you plotting $\displaystyle (\cos t,\sin t)$ to get a 'unit circle'.

So, let us give some sine and cosine definitions using the unit circle and then using right triangles.

Unit Circle Definition

this is a placeholder image. It needs to be redrawn with x, y and the angle shown - and nicer colours.

If a line of length 1 is drawn at an angle of $\displaystyle \theta$ to the x axis (where the angle is measured in an anticlockwise direction from the x axis) then the $\displaystyle x$ coordinate is given by:

$\displaystyle x = \cos \theta$

and the $\displaystyle y$ coordinate is given by:

$\displaystyle y = \sin \theta$

Notation and pronunciation $\displaystyle \cos$ is of course just an abbreviation for 'cosine', and $\displaystyle \sin$ is just an abbreviation for sine. Rather confusingly $\displaystyle \cos$ can be pronounced either 'cos' or 'coz' always with 'o' as in 'bottle', rather than 'o' as in 'code' and $\displaystyle \sin$ is often pronounced 'sine' rather than 'sin'. It's not very logical, it's just how it is.

Ratios of Sides Definition

The sides in a right triangle can be labelled like so:

this is a placeholder image, we'll re-do it with an angle clearly shown.

• We already know that the long side is called the hypotenuse.
• The sides next to the angle we have chosen is called the adjacent side.
• The remaining side which is opposite the angle is called the opposite side.

The angle determines the ratios of the side. Once the angle is selected we can make the whole triangle larger or smaller but all lengths change in the same proportions. We can't change the length of one side without also changing the length of all sides in the same proportion, or else we have changed the angles. So, once we know the angle we know the ratio of the sides. The functions that give us those ratios are defined as:

$\displaystyle \frac{\text{opposite}}{\text{hypotenuse}}=\sin \theta$

and

$\displaystyle \frac{\text{adjacent}}{\text{hypotenuse}}=\cos \theta$

'Unit Hypotenuse' Definition

This definition of sine and cosine isn't usually given, but it is also valid.

Choose a triangle with a hypotenuse of unit length and label it like so:

this is a placeholder image, we'll re-do it with an angle clearly shown and unit hypotenuse

Then:

$\displaystyle \text{opposite}=\sin \theta$

$\displaystyle \text{adjacent}=\cos \theta$

Exercises

Exercise: These definitions amount to the same thing Use this third definition to convince yourself that the three different ways of defining sine and cosine amount to the same thing, at least for angles between 0o and 90o.

Exercise: Unit Circle Did you do the exercise on Plotting (Cos t, Sin t) on the previous page? It really is important to have had a go and seen how cosine and sine are related to the unit circle. If nothing else you MUST be able to use $\displaystyle \cos$ and $\displaystyle \sin$ on your calculator or you will not get very far with trigonometry.

Exercise: To think about The unit circle definition of the trig functions shows that we can work with angles greater than 90o. 90o represents a quarter of a circle. 360o represents a complete circle. What happens, or what should happen for $\displaystyle \cos$ and $\displaystyle \sin$ if we have angles greater than 360o?

Tangent

There is one more trigonometric function that we want to introduce on this page. It's the tangent function or just $\displaystyle\tan$.

For the unit circle definition we define the tangent of theta as:

$\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta}$

For the ratios of sides definition we define the tangent of theta as:

$\displaystyle \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Using the definition of sine and cosine in terms of a triangle with unit hypotenuse it is immediately clear that these are the same thing.

These definitions of Tan amount to the same thing If we didn't have the definition of sine and cosine in terms of the triangle with unit hypotenuse we'd need to do slightly more work to show that the two definitions of tan were equivalent. We'd do something like this: $\displaystyle \tan \theta = \frac{\sin \theta}{\cos \theta} = \overbrace{ \frac{\sin \theta}{1} \times \frac{1}{\cos \theta} = \frac{\text{opposite}}{\text{hypotenuse}} \times \frac{\text{hypotenuse}}{\text{adjacent}} }^{\text{Using definitions of} \sin \text{and} \cos \text{as ratios}} = \frac{\text{opposite}}{\text{adjacent}}$ It is worth checking every step in this.

Tan or Tangent? When talking about the tangent function $\displaystyle\tan$ it is usually better to always just say 'Tan' rather than 'Tangent'. The reason is that 'Tangent' also has another meaning in mathematics. A 'tangent' to a circle is a straight line that touches the circle but does not cross it - a tangent to a circle, even when extended as far as you like in both directions meets a circle at just one point. The line from the one point where it meets the circle to the centre of the circle is always at right angles to the tangent line.