Trigonometry

… Sin, cos and tan can be thought of as ‘sausage machines’ that give you a ratio of two sides of a right angled triangle if you ‘put in’ an angle (this is the SOH CAH TOA method).

… Sin and cos curves are similar, but:

y = sin(x) starts at (0,0) and looks like a backwards ‘S’ on its side.

and

y = cos(x) starts at (0,1) and looks like a ‘C’ on its side.

… y = tan(x) is a completely different curve with ‘asymptotes’ where the curve goes to infinity! It repeats every 180 degrees.

It is useful to rule light vertical lines at 90 and 270 to show where the tan graph goes to positive or negative infinity

… To find sin, cos and tan of 30 or 60 degrees without a calculator, use an equilateral triangle with sides of length 2, but split down the middle to make a right angled triangle of height √3.

… To calculate sin, cos and tan of 45 degrees without a calculator, use a 45° triangle with side lengths 1 and hypotenuse length √2.

… You’ll need to learn that:

sin(0)=0 sin(30)=½ sin(45)=√2/2 sin(60)=(√3)/2 sin(90)=1

cos(0)=1 cos(30)=(√3)/2 cos(45)=√2/2 cos(60)=½ cos(90)=0

tan(0)=0 tan(30)=1/√3 tan(45)=1 tan(60)=√3 tan(90)=∞ (infinity)

Here’s a video which shows you a hand trick you can use to remember the values for sine and cosine: https://youtu.be/xXGfp9PKdXM

Remember that your thumb starts at 0° and the fingers continue 30°, 45°, 60° and 90°. Your thumb side is always ‘sin’.

Whatever number you get, square root it and divide by 2.

Trigonometry method for calculating lengths and angles:

… You’ll know when you need to use SOH CAH TOA trigonometry because the question will include:

1) a right angled triangle(s), and

2) length(s) AND an angle

Step 1: LABEL your right-angled triangle with Opposite, Hypoteneuse and Adjacent and CROSS OUT the side that you are not interested in.

Step 2: select the correct equation from SOH CAH TOA and write the formula – no numbers yet! Call the angle ‘theta’ (egg with bandana!).

Step 3: SUB-IN the numbers and solve (rearrange as needed, changing the subject of the equation)

For finding angles, use exactly the same method as above. In Step 3 you’ll need to feed the ratio into the inverse trigonometry function (‘shift’ then sin, cos or tan) to get the angle.

So if sin(A) = 0.8

Then the angle A = inverse sin ( 0.8 )

Note that inverse sin here looks like it is “sin to the power of -1” but it is not a power… The -1 just means ‘inverse’

Further trigonometry

… The cosine and sin rules are for finding lengths and angles in non-right angled triangles.

… When to use the cosine rule: at the end of the question you will have THREE lengths. If not, then it’s usually the sin rule.

… When using the cosine rule, always make the length you are calculating is the ‘a’ side, or the angle you are calculating the ‘A’ angle.

… It’s essential to label the triangle first (depending on the question, you might need to re-label!)

… The question will often suggest labels for A, B and C… However – ignore these and label the triangle yourself (sometimes their labelling is wrong and can put you off track!)

… If an angle is described as ABC it means that the angle is located at letter B (the middle letter).

… Calling ‘x’ the length that you want to find, or calling ‘theta’ the angle that you want to find will help you to remember whether you are dealing with a length or an angle :).

… You can use the sine rule in both the right way up, and upsidedown forms, depending if you want to find an angle or a length.

… Sometimes you may need to use ‘sum of angles in a triangle add to 180°’ to find another angle, or use the geometry of the situation to find as much information as possible before using the sine or cosine rule

… Area of a non-right angled triangle

A = 0.5absin(C).

You’ll need a side-angle-side triangle to use this.

3D Trigonometry and Pythagoras

… First draw in any lines suggested by the question and add any information given in the question to your diagram.

… Try to identify the right angled triangle in the 3D shape and then draw it out in 2d.

… For pyramids, it often helps to draw extra diagonal lines from each of the base’s corners and then a vertical line from the base mid point to the top vertex. You may need to find the length of the base diagonal first (the hypotenuse) and then halve it.

… A diagonal line on the base is also useful when finding the angle between the sloping face of a ‘wedge’ shape and its base.

… Remember to write out the formula once you have labelled the triangle, and then put in the numbers (to gain the method marks)

… To avoid rounding errors during the calculation, either work in surd form (for example, “4√2”, or to one more significant figure than is ruined for the final answer.

… Look out for keywords in the question such as ‘SQUARE based pyramid’ which will help you to find other sides in the diagram.