Once we know the addition formulas, the subtraction formulas are a cinch. We just plug in a negative angle and watch the signs.
sin (α – β) = sin (α + (-β))
= sin (α) cos (-β) + cos (α) sin (-β)
Don't be such Negative Nancy's, sine and cosine. We'll use our knowledge of periodicity to pull those negative signs out of the trig functions.
Cosine is an even function, so a negative angle doesn't affect the sign. cos (-x) = cos (x)
Sine is an odd function, so a negative sign is pulled out to the whole function. sin (-x) = -sin (x)
sin (α – β) = sin (α) cos (β) – cos (α) sin (β)
This formula is just like the addition formula, except the sign in the middle is switched around. That is mighty convenient and easy to remember. Wouldn't it be nice if cosine and tangent worked the same way?
cos (α – β) = cos (α) cos (-β) – sin (α) sin (-β)
cos (α – β) = cos (α) cos (β) + sin (α) sin (β)
Oh?
Oh! They do work the same way. The middle term switches sign each time. That is nice.
Sample Problem
Find the exact value of cos (15°).
With 15° being so small, it should be pretty obvious that we need a subtraction formula for this one. We could do 15° = 60° – 45° or 15° = 45° – 30°. Either one would work, and neither one would be faster than the other. Let's work with *flips coin* 45° – 30°.
cos (45 – 30) = cos (45) cos (30) + sin (45) sin(30)
We wish we felt half as special as these angles do.
And now we can add and subtract angles with ease. Well, as long as we know the formulas.