Erm, questione about the Reduction Formula (trig). Thanks!!!?

Question

HI! I'm pissed, as usual, about trigonometry. This time, it's about the reduction formula, a.k.a. the most idiotic thing I've ever seen in my life. I thought that all was fine, but now I've come across some real problems. First of all, my book does not explain how to use the reduction formula. All it does is give the formula and then show 1 problem worked out.

What I've surmised from that (and it has served me well thus far) is:

√(a² + b²)sin( x + α)

Great. I get the whole "√(a² + b²)sin( x" part, but I seem to be having trouble with the "+ α" aspect of it.

What I would guess, solely from my stupid book, is that to get α, you find cos α by doing a/[√(a² + b²)]. I need to know:

1) How do you know which value of α to use? Often times there is more than one angle for which cos = a/[√(a² + b²)].

2) What do you do when the problem is NOT in the form of a sin x + b cos x, but rather a sin x - b cos x? What about if a is negative?

Thank you so much! I'll pick a best answr!

Answer 1

The basic reduction formulas may be summarized as follows:
n times 90 degrees +/- α =, where n is any even integer or 0, is numerically = to the same function of α. Any function of n times 90 degrees + α, where n is any odd integer is numerically = to the corresponding cofunction of α. The algebraic sign of the result is the same as the sign of the given function in that quadrant in which n times 90 +/- α would lie if α were an acute angle.

sin (270-α) = cos α since 270 is an odd multiple of 90. If α were acute , 270-α would lie in the 3rd quadrant where the sine is negative. Hence sin(270-α) = -cos α.

Your reduction formula is Acos x + Bsin x = Ccos(x-D). This is where C is the square root of A^2 + B^2 and cos(D) = A/C and sin(D) = B/C.
Hint: Draw a reference triangle. Calculate the angle. Determine which quadrant the angle is located by comparing the signs of A and B. If (A,B) is: (+,+) = I, (-,+) = II, (-,-) = III, and (+,-) = IV. Place those values into the formula above. An example of this would be: 2cos x - 2sin x. C^2 = 4 + 4= 8, C = 2(2)^½. The quadrant with a negative sinx and positve cosx is the fourth quadrant. Thus the expression simplifies to
2(2)^½cos(x- 315°).

Answer 2

What thou loookest for thou won't locate (and please warms those arms first in case you do no longer suggestions) My suggestion - take decrease lunch while "going bush" this could take a protracted whilst so relax your tush this is not any longer a good flow in scheme of romance yet I see out of your face you may take a raffle score me out of 10 once you get the time it could motivate me to place in writing you a rhyme