Factor Formulae

This page explores factor formulae. The factor formulae can be used to express sums and differences of sines and cosines as products of sines and/or cosines. The factor formulae shown below are derived from the addition formulae.
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In the following example we’re going to use the formulae sin(A+B) and sin(A-B) to derive the result that;
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We know that;
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…adding the two identities gives;
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First let A+B=P and A-B=Q therefore;
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…therefore…
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Example

In this example we’re going to use the following factor formulae;
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First we shall show that;
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Let 105° and Q=15° …so now we have…
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…that gives…
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Next we shall solve for 0≤θ≤π, sin4θ – sin3θ = 0
Using the factor formulae above let P=4θ and Q=3θ, so we have;
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We’re trying to find the solutions for;
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The method is very similar to how to solve quadratic equations, first let us start with;
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so;
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…therefore…
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Next we solve for;
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…we find that θ=0.
The final solutions are;
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Example

Here is another example worth going through. Suppose we had to prove that;
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This is actually very simple to prove. Let us start in the LHS and try to find tan(x + y). Let us the factor formulae for the numerator. We shall use;
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…we shall set;
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For the numerator we have;
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…so now we have…
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We can do the same for the denominator;
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Now we can use both equations;
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…that leaves…
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