Maths for science and technology

6 Basic trigonometry

We are going to look at some of the basics of trigonometry relating to right angle triangles. So the first question is, What is a right angle triangle?

It is a triangle in which one of the angles is 90°, which is commonly referred to as a right angle. The sum of the angles in any triangle is 180°. So if the other two angles are α and β as in Figure 1

Look at the angle β in Figure 1. The side of the triangle opposite to this is labelled b and is called the ‘opposite?. The side next to β is labelled a and is referred to as the ‘adjacent?. The side labelled c is the called the ‘hypotenuse?. The hypotenuse is always the side opposite the right angle.

Pythagoras' Theorem states that the square of the length of the hypotenuse of a right angle triangle is equal to the sum of the squares of the lengths of the other two sides. Looking at the triangle in Figure 1

This formula, together with some knowledge of trigonometry enables us to calculate the angles and sides of the triangle in Figure 1: (given some other information).

Now look at your calculator. A scientific calculator is essential if you are going to use any trigonometry. Scientific calculators operate in two modes when dealing with angles. Before embarking on any of the examples please ensure that your calculator is in degree mode. Follow the instructions in your calculator manual to do this.

You will see three buttons, probably close to the screen and to the right hand side of the keypad, labelled sin, cos and tan. These are the functions we will be using and are called trigonometric (or ‘trig?) functions.

We can now define these functions for you, based on the angle β and the right angle triangle in Figure 1.

There is an easy way to remember this: SOHCAHTOA.

If, for any right angle triangle, we know two sides or a side and an angle, then we can find the other side, sides or angle as required.

You might like to experiment with your calculator finding sin and cos for various angles. You should notice that sin and cos always lie between 0 and 1 if the input angle is between 0° and 90°.

Example 7

We are going to find the unknown values, α, a and c in Figure 2.

Discussion

In Figure 2, the angle β is 30° and the side b is 6 cm.

The unknown angle α can be found using the fact that the sum of the internal angles of a triangle is 180°.

Now calculate a, which is adjacent to the angle β, which is 30°. We know the opposite side

is 6 cm.

If you look at SOHCAHTOA, you will find that the function relating adjacent and opposite is tan.

The side α is 10.392304485 cm.

The only remaining unknown is c and this we can find using Pythagoras? Theorem.

The side c is 12 cm.

Using the rules for a triangle, some basic trigonometry and Pythagoras, and knowing the length of one side and the size of one angle, we have found the remaining angles and sides of the triangle.

Activity 15

Now try this one yourself. Find the unknown side b and the angles α and β of Figure 3.

Answer

Discussion

This time we know the length of two sides. In relation to the angle β, they are the adjacent and the hypotenuse.

The trig function which relates these two sides is cos.

This tells us that β is the angle for which cos is 0.6; this can be written β = cos⁻¹ 0.6. We can now use a calculator to find β.Look at your calculator again. You should see a key which has the letters inv above it. This key enables you to use the functions written above, rather than on the other keys. Look at your sin, cos and tan keys again. Above each you should see what are known as the inverse functions, sin⁻¹, cos⁻¹ and tan⁻¹.

To find cos⁻¹ 0.6, key in 0.6, press inv and then the cos key. The value which appears should be 53.13010236.

(Some calculators may require a slightly different methodology, so please check your calculator handbook.)

So, β = 53.13° (to 2 decimal places)

The angle α can be found in the same way as the third angle in the example. α = 180 − (90 + 53.13) = 36.87°.

The opposite side b can be found using Pythagoras.

The length of side b is 4 cm.

Here are two more exercises for you to try.

Activity 16

In the Figure 4, find the angles β and α and the length of side a.

Answer

Discussion

and, since all the angles inside a triangle must add to 180°

From Pythagoras

The length of side a = 6.24 cm (to 2 decimal places).

So we know all the sides and all the angles in the triangle.

Activity 17

Look at Figure 5 and find the sides a and b and the angle α.

Answer

Discussion

This time we are starting with different information, but we still need to look for two known values related to one of the unknown values.

and

To find the other angle

This is a particular type of triangle where side a = side b and there are two equal angles.

This sort of triangle is known as an isosceles triangle.

What is the point of trigonometry?

One application that you might come across in applied mathematics, physics or technology courses concerns the resolution of forces.

Consider the case of a father pulling his two young children along on a sledge (Figure 6a). He has a rope attached to the sledge which makes an angle α with the ground. The force F in the rope is 50 N. If α is 25°, find the horizontal and vertical components of the force.

The force F acting in the rope may be considered as two component forces. We will call the horizontal component F_h and the vertical component F_v (Figure 6b).

Looking at Figure 6b and applying the trigonometry from earlier in this section, we can say

and

Rearranging these two equations gives

and

The horizontal component of the force is 45.32 N.

The vertical component of the force is 21.13 N.

We should find that F_h² = F_v² = F²

Check using your calculator.

Now here is an activity for you.

Activity 18

Calculate the horizontal and vertical components of the force F if

1. the angle α is reduced to 15°

2. the angle α is increased to 45°

Answer

Discussion

(a)

The horizontal component of the force is 48.30 N.

The vertical component of the force is 12.94 N.42

(b)

The horizontal component of the force is 35.36 N.

The vertical component of the force is 35.36 N.

You can check these values using Pythagoras.

(a) 48.30² + 12.94² = 50²

and

(b) 35.36² + 35.36² = 50²

Now look at these values.

Can you identify any relationship between the angle and the components of the forces?

As the angle α increases, the horizontal component of the force decreases and the vertical component increases. In order to maximize the effectiveness of the force in terms of horizontal motion, the angle α should be kept as small as possible.

The activities in this section have been designed to help you to use trigonometry in problem solving. As you continue your studies you will encounter other different applications.

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