A trigo question need help!?

Question

Let f(x)=3sin(x)-2cos(x). show that function f can be written as a sine function which has been shifted horizontally, then stretched vertically. Find the amplitude of function f.

Answer 1

Using this:
http://en.wikipedia.org/wiki/Trigonometry#Sum_and_difference_identities
And more particularly this:
http://mathworld.wolfram.com/HarmonicAdditionTheorem.html

We know that f(x) =

c sin (x + β)

c = √(3² + (-2)² )

β = arctan( - 3 / (-2) = arctan(3/2)

You don't really need β, all you want is c, the amplitude of f. So compute:

√(3² + (-2)² )

= √(9+4) = √(13)


For further explanation/discussion see:
http://cda.morris.umn.edu/~mcquarrb/Precalculus/Resources/HTMLLinks/Lecture4.6_4.html
For an interactive applet that computes and graphs the sum of two (or even three) sin functions, see:
http://www.analyzemath.com/trigonometry/Add_Trig_Functions.html

Answer 2

Let
3 sin x - 2 cos x = k sin (x - a)----k>0
= (k cos a) sin x - (k sin a) cos x

3 = k cos a
2 = k sin a--------angle a in 1st quadrant

3² + 2² = k ²(cos ² a + sin ² a)
13 = k ²
k = √13

2 / 3 = tan a
a = 33.7 °

f (x) = √13 sin (x - 33.7)°