Question

Use appropriate identities to rewrite the wave equation shown below in the form ℎ(x) = a cos (x − c).

Answer 1

Answer:

$h(x) = 10 cos(x-36.870)$

Step-by-step explanation:

Note: For this problem we use the calculator on degrees

For this case we need to remember this identity :

$Cos (a-b) = cos a cos b + sin a sin b$

For this case if we apply for our desired formula we got this:

$a cos (x-c) = a [cos (c) cos (x) + sin (c) sin (x)]$

And we want this equal to $h(x) = 6 sin (x) + 8 cos (x)$ so we can set up the following equality:

$6 sin (x) + 8 cos (x)= a cos (c) [cos (x)] + a sin (c) [sin(x)]$   (1)

If we apply direct comparison between the factors on equation (1) we see this:

$a cos(c) = 8$    (2)

$a sin (c) = 6$    (3)

If we solve a from equation (2) we got:

$a = \frac{8}{cos (c)}$   (4)

If we replace equation (4) into equation (3) we got:

$\frac{8}{sin(c)} cos (c) = 8 tan (c) = 6$

$tan(c) = \frac{6}{8}=\frac{3}{4}$

If we apply inverse tangent on both sides we got:

$c = tan^{-1} (3/4) = 36.870$

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:

$[tex] a = \frac{8}{cos (36.870)}=10$

And then our expression in the form $h(x) = a cos (x-c)$ is:

$h(x) = 10 cos(x-36.870)$

And we can check that:

$h(x)= 10 cos (36.870) [cos (x)] + 10 sin (36.870) [sin(x)]= 8 cos (x) + 6 sin (x)$