Trigonometry/Remembering Trig Formulae
This page is about how to remember trig formulas - so it is more than just a summary of trig formulas.
You will need to have read most of the rest of the book for this page to be useful to you.
The Blue-Box Formulae
Tips on checking and remembering Formulae
This page contains some trigonometric identities. It does not contain all trigonometric identities. It couldn't possibly. As well as the trigonometric identities for 'double angles' such as:
there exist trigonometrical identities for treble angles, quadruple angles and so on. We could not list them all. Also the same formula can be presented in different disguises as we'll see below.
It's useful to learn some of the trig identities and to know how to quickly and easily derive one trigonometric identities from another.
It is also very important to be able to check that the formulas you come up with make sense. We know that . If your formula for does not give the answer 1 when then it is wrong! Don't stop there though. You can work out where you went wrong and save yourself making the same mistake again. How do you do that?
If you worked out the formula for from the formula for you need to check through the steps you took. If the mistake isn't obvious, try putting A=0 and B=0. It is very very easy to get a sign wrong, a plus for a minus, when working with equations quickly. It is only through practice that you will become both fast at it and accurate.
Identities Based on the Pythagorean Theorem
Some of the most fundamental trigonometric identities are those derived from the Pythagorean Theorem. These are defined using a right triangle:
At this stage in the course the Pythagorean Theorem should be second nature to you. If you have not yet learned it, learn it now.
and of course are the legs or the adjacent and opposite edges, and is the hypotenuse, the longest side, the side that does not include the right angle. This formula only works for a right angle triangle. If the angle shown as a right angle in the diagram were obtuse, larger than a right angle, then would be larger than the Pythagorean sum. If the angle shown as a right angle were smaller than a right angle, then would be smaller than
Pythagorean Theorem in Terms of sine and cosine
The Pythagorean Theorem is the same thing as
We know this from the earlier in the book where we introduced and .
- The is the adjacent side when the hypotenuse is one.
- The is the opposite side when the hypotenuse is one.
The identity should be second nature to you too, but you should also be able to derive it from the Pythagorean relation.
We can see it's true from a right triangle that has a hypotenuse that is 'c' rather than one. Dividing the Pythagorean relation through by c2 gives us
And we have already seen from soh-cah-toa that the sine of A is and the cosine of A is .
Identities based on Addition Formula
We saw the addition formula earlier.
Letters are Arbitrary
The letters in these equations and are arbitrary. We could equally use and . We're pointing this out because we happen to have a diagram on this page that has angles , and in it, and we want to make it clear that that was for Pythagoras and we're not talking about that diagram any more. In these equations you can also always substitute actual values, e.g. replace by and they'd still be true. Or you could replace by or and they'd still be true.
The addition formula for cosine is:
and is worth learning. But don't learn the next one:
Learning the second formula is extra effort that does not really gain you anything. You can get it instantly by replacing by in the first equation. The only things on the right hand side that will change are the terms that had B in them. The is unchanged because . The minus sign before the second term changes because .
Checking the 'subtraction' formula
Now do a quick check. If then is which is one, right? And on the right hand side we have , which is also one. Looks good.
It is much better to remember it this way, because you are doing less rote learning and you're also becoming more fluent with the algebra.
Double Angle Formula
You can immediately get to the next formula by putting B equal to A in the addition formula. This is a double angle formula.
Half Angle Formula
Replace A by A/2 in the previous result and you get the half angle formula.
Sum of two shifted Cosines
For the next formula, you could just plough through the algebra...
Waves in and out of phase
Before we do, notice that we are adding two cosine waves, the first shifted left and the second shifted right. We've seen this before when we looked at waves in and out of phase. We're going to get another sinusoidal wave. The expression is fairly symmetric, and actually we can quickly show that the expression gives an even function, that is that the value for -x is the same as for x:
- using cos(x) = cos(-x) we get to:
- and now swapping the two terms:
- which is the original expression.
with practice you'll be able to see those steps and that the expression is an even function immediately.
A sinusoidal wave that is an even function - well it is something based on cosine. We're expecting the formula will simplify to give us something like:
Where A will depend on .
Values to Check With
When (try it in the original formula) we expect A to be 2. When we expect A to be 0, as the two cosine waves are out of phase. So now to plough through the algebra:
And to check we try with and get , and we check with and get , as we hoped.
Knowing roughly what we expected made it easier to get the algebraic steps right. We knew where we were going. We knew we weren't going to end up with four separate terms, some would have to cancel or combine in some other way.
Identities based on Sine and Cosine rules
This is easy to remember because it is so symmetrical. You do not even need to remember which way up the ratios are as this is also true:
What you do need to remember is how the triangle is labelled for it to be true.
With the cosine law formula:
It is best to think of it as a more general version of Pythagoras' theorem. The a and b have to be on equal footing, so ab as a multiplier is reasonable.
- If you are familiar with 'units' in physics, then use the fact that units must match up. The quantities a,b,c are measurements of length. It is no good adding to . One might have units Km2 the other units of Km. Adding to the units, e.g: Km2, do match.
In some more detail, it has got to be minus for . That's because for we need to be less than for Pythagoras' theorem.
How can you remember and be sure the 2 is right? Think of an equilateral triangle with each side of length 1. The angle and . We need the 2 to get the right answer 1 + 1 + 2x1x1x0.5 = 1.
This is one definition of tan, and is something you should just learn.
Identities based on symmetry
These symmetry identities are best remembered by remembering the graphs for these functions. Have a look at the graphs again if you don't remember them.
are landmarks on the graphs of sine and cosine, and you should know what happens for both those functions at those landmarks. When you have that knowledge the following symmetry identities are easy to write down, because you can see them visually.
The identities above are visualised as shifting the graph left by . It's a half cycle, and both cosine and sine change signs - and hence tan does not change sign, because when it is expressed as a ratio of sine over cosine both the top and bottom change signs.
Reflecting in x=π/2
In the above we are reflecting the graph about the vertical line . Spend enough time looking at this on the graph to see that that is true. It is important to do that to remember this trick for remembering what happens.
Now sine has a maximum at (it has the value one) and it is symmetric about that x value, so . Cosine is zero at . Reflecting cosine in the line reverses the sign.
The above two formulas are most easily seen from the right-triangle definition of cosine and sine. The two acute angles in such a triangle add up to , and we are just giving the ratios of sides lengths in terms of a different angle's trig function.
These two formulas are 'the same' as the previous formulas. We get there in two steps. First subtracted pi from the angle on the right hand side, which inverts the sign, and inverted the sign to compensate. Next invert the sign of the angle on the right hand side. For cosine on the right we are done. For sine on the right we have to invert its sign since inverting the angle's sign changed the sign of the result.
Yes - keeping track of signs is tricky, and always needs care.