GCSE Mathematics/Trigonometry

Pythagoras

A use for pythagoras' theorem is to find the side of a hypotenuse of a right angled triangle, that being, the longest side. To do this, we can use the formula:

The sides of a right triangle

$a^{2}+b^{2}=c^{2}$

c being the hypotenuse and longest side, while a and b being the two other sides.

Then, to find the length of the hypotenuse, we must squareroot the result.

How does it work?

The therom says the two sides squared. This literally means squared, for example drawing a square with the lengths the same as a. The area of two squares, one with all lengths equal to a, and one with all equal to b, would equal the area of the square of the hypotenuse:

The squares of the sides

Due to this we can also do it in reverse. The hypotenuse, minus one of the other sides will equal the square of the third and final side.

$c^{2}-b^{2}=a^{2}$ or $c^{2}-a^{2}=b^{2}$

Examples

An example of use of pythagoras would be in a triangle which has sides:

• a = 4 cm
• b = 6 cm

We must do

$4^{2}+6^{2}=52$

Then square root as 52 is the square of the side.

√52 = 7.211

Therefore, the other side is 7.211

SohCahToa in right triangles

We can work out missing sides in a *Right triangle* using one angle and one side, or an angle using two sides. This can be done using the sin, cosine and tangent functions.

There are three sides in any triangle. In a right angled one, the side opposite the right angle is the hypotenuse, the one opposite the given angle/the angle being found out is the opposite, and the last one the adjacent.

The sides have been labelled in this triangle in relation to angle A.

Area of triangle with sine

Use the formula

1/2 bcSinA

1/2 abSinB

1/2 acSinA

Sine rule

For finding sides use

  a/sinA =b/sinB =c/sinC

For angles finding angles

  sinA/a =sinB/b =sinC/c