Trigonometric Addition and Difference Formulas (Identities) Also double angle formulas.


The addition and difference formulas used in trigonometry are as follows:

sin (A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB – cosAsinB

cos(A+B) = cosAcosB – sinAsinB

cos(A-B) = cosAcosB + sinAsinB


tan(A+B) = (tanA + tanB) ÷ (1 – tanAtanB)

tan(A-B) = (tanA - tanB) ÷ (1 + tanAtanB)

These formulas can be helpful in a variety of situations, one being in writing down exact values of sinA, cosA or tanA, and also deriving other trigonometric formulas such as the double angle formulas.

Example 1

Work out an exact solution to sin15⁰.

If you change 15 to 45 – 30, then the second trigonometric addition formula from the list above can be used to work out the exact solution to sin15⁰:

sin(A-B) = sinAcosB – cosAsinB

sin(45-30) = sin45cos30-cos45sin30

Now the values of sin45, cos30, cos45 and sin30 should already be known. If you don’t know them then these can be worked out by drawing out some right angle triangle. These are:

sin45 = (1/√2)

cos30 = (√3/2)

cos45 = (1/√2)

sin30 = ½

So:

sin(45-30) = (1/√2)× (√3/2)- (1/√2) ½

= (√3/2√2) – (1/2√2)

= (√3 -1)/2√2

Let’s take a look at a slightly different example of using these addition/difference trigonometric identities:

Example 2

Write cos6xcosx + sin6xsinx as a single trigonometric function.

Here A = 6x and B =x, so all you need to do is substitute these values into the 4th formula listed at the top of this page:

cosAcosB + sinAsinB

= cos6xcosx + sin 6xsinx

= cos(6x-x)

= cos5x

You can also work out the double angle formula in a similar way, you can do this by changing the B’s to A’s in the 1st, 3rd and 5th formulas. Let me demonstrate this in the first trigonometric addition formula as the other double angle formulas will be covered in a later hubpage:

sin(A+B) = sinAcosB + cosAsinB

Now, change all of the B’s to A’s in the formula above:

sin(A +A) = sinAcosA + cosAsinA

Next, you can simplify this expression to give:

Sin(2A) = 2sinAcosA

This is known as the double angle formula for sin2A.

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