Question #0a796

1 Answer
Feb 17, 2018

Answer:

#3.7 cos(pi t-5.953)# i.e. the constants are #A = 3.7#, #omega = pi# and # phi =2pi-tan^{-1}(12/35)=5.953 #

Explanation:

To express the given function in the form #A cos (omega t + phi)#, notice that according to the trigonometric sum rule

#A cos (omega t + phi) = A [cos (omega t )cos (phi)-sin(omega t)sin(phi)] = -A sin(phi) sin(omega t) + A cos(phi)cos(omega t)#

Comparing with the expression given, we have #omega = pi# nd

#A sin(phi) = -1.2, qquad A cos(phi) = 3.5#

Thus

#A^2 = = (A sin(phi))^2 + (A cos(phi))^2= (-1.2)^2+(3.5)^2 = 13.69 = 3.7^2#

So, #A = 3.7#

Now,
#tan(phi) = {A sin(phi)}/{A cos(phi)} = -{1.2}/{3.5} = -0.3429#

Since #sin(phi)# is negative while #cos(phi)# is positive, the angle #phi# must be in the fourth quadrant. So

#phi =2pi-tan^{-1}(12/35)=5.953 #