Yesterday in class we learned equivalent trigonometric expressions, which are co-function identities for angles in the 1st and 2nd quadrants. For all of the 12 identities, the angle x is either added or subtracted from p/2 or 90°. Today in class though we learned what to do when x isn’t added or subtracted from p/2 , and the formulas we use are called compound angle formulas.
Ms. Burchat showed us how the cosine compound angle formulas were derived, in order to prove that they worked. In summary, on a unit circle the coordinates of a point relevant to the angle is (cosx, sinx) with x being the angle. Two points, a and b, were plotted on the unit circle and the difference in their angles(a-b) is what we were trying to solve. After a lengthy process, the formula that was arrived to is
Cos(a-b) =cosacosb + sinasinb
Ok, on to some examples.
The first example we did was cos (p/3 - p/4).
Before we start, let’s see what the decimal answer is when u punch the angle into your calculator. In degree measure, the statement is cos(60°-45°) which equals to cos(15°) . In your calculator, this value is 0.965925826. But we know that this is far to big a number, so we would resort to rounding the value to approximately 0.9659. But this is not the most accurate statement; there must be another way! This is where the compound angle formulas come into play.
Since this is cos(a-b) we will use the formula Cos(a-b) =cosacosb + sinasinb.
Cos (p/3 - p/4) = cosp/3 cosp/4 + sinp/3 sinp/4
= (1/2) (1/√ 2) + (√ 3/2) (1/√ 2)
= 1/2√ 2 + √ 3/2√ 2
= 1+ √ 3/2√ 2.
This value is the most precise measurement for the angle and the most accurate.
The other examples done in class were similar but they just used one of the different formulas. Hope this is explanatory!