Respuesta :
Answer:
[tex] h(x) = 10 cos(x-36.870) [/tex]
Step-by-step explanation:
Note: For this problem we use the calculator on degrees
For this case we need to remember this identity :
[tex] Cos (a-b) = cos a cos b + sin a sin b[/tex]
For this case if we apply for our desired formula we got this:
[tex] a cos (x-c) = a [cos (c) cos (x) + sin (c) sin (x)][/tex]
And we want this equal to [tex] h(x) = 6 sin (x) + 8 cos (x)[/tex] so we can set up the following equality:
[tex] 6 sin (x) + 8 cos (x)= a cos (c) [cos (x)] + a sin (c) [sin(x)][/tex] (1)
If we apply direct comparison between the factors on equation (1) we see this:
[tex] a cos(c) = 8[/tex] (2)
[tex] a sin (c) = 6[/tex] (3)
If we solve a from equation (2) we got:
[tex] a = \frac{8}{cos (c)}[/tex] (4)
If we replace equation (4) into equation (3) we got:
[tex] \frac{8}{sin(c)} cos (c) = 8 tan (c) = 6[/tex]
[tex] tan(c) = \frac{6}{8}=\frac{3}{4}[/tex]
If we apply inverse tangent on both sides we got:
[tex] c = tan^{-1} (3/4) = 36.870 [/tex]
So then the value of c= 36.870 degrees. And since w ehave the value of c we can find the value for a and we got:
[tex] [tex] a = \frac{8}{cos (36.870)}=10[/tex]
And then our expression in the form [tex] h(x) = a cos (x-c) [/tex] is:
[tex] h(x) = 10 cos(x-36.870) [/tex]
And we can check that:
[tex] h(x)= 10 cos (36.870) [cos (x)] + 10 sin (36.870) [sin(x)]= 8 cos (x) + 6 sin (x)[/tex]
