# Neat Trick for remembering the sines and cosines of 0,30,45,60 and 90

Dear H,

We have just spent 2 lessons on Trigonometry and here is the writeup as promised. You were getting everything right but these notes are in case you forget a small step and need to revise.

Sorry you couldn’t remember SOH CAH TOA, but it’s really quite complicated, what with labelling the triangle, writing out SOH CAH TOA, remembering which way round they go, what they mean and how to use them…

So anyway we were just using the SINE RULE and CROSS SUMS to find the missing side or the missing angle.

Before you start, make sure you have revised

- Cross Sums
- Pythagoras’ Theorem

- Work out the value of the missing angle and write it into the correct corner. (For example, the 3rd angle in a 90/42/? triangle would be found by doing 180-90-42=)
- Draw a cross sum. In the top line, write the KNOWN SIDE and an x (or ?, or whatever the exam is calling the side you are looking for. In the images below we labelled it “?”)
- Underneath the KNOWN SIDE write down the sin of its opposite angle
- Underneath the “?” side, write the sine of its angle. That’s sin(?)
- Write down the sum you plan to type into your calculator
- And do it!
- Write the answer back on the diagram and check it looks sensible

- If the Hypotenuse is unknown you will have to use PYTHAGORAS’ Theorem to find it, first.
- Draw a cross sum. In the top line, write the KNOWN SIDES and an x (or ?, or whatever the exam is calling the side you are looking for. In the images below we labelled it “?”)
- underneath the known sides, write sin(of their opposite angle), one should be sin(90), the other will be sin(?)
- Write down the sum you plan to type into your calculator
- This time the cross sum’s answer will be weird (0.67834 in the example below) and you need to write sin(?)= 0.67834 or whatever.
- You found the sine of the angle – to find the angle itself, press Shift, Sin, ANS, =
- Write the answer onto the diagram and check it looks reasonable.

If the hypotenuse and one other side is given, then you wont need Pythagoras. So its easier.

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