Express the function y(t)= 4 sin 2πt + 15 cos 2πt in terms of (a) a sine term only. (b) Determine the amplitude, the period, the frequency in hertz of the function. (c) Draw the function in time domain.

y(t) = 15.52sin(2πt + 75.06°)

Amplitude = A = 15.52

Frequency = f = 1 Hz

Period =  1 s

Step-by-step explanation:

The given function is

y(t)= 4sin2πt + 15cos2πt

(a) a sine term only

we can use phasor technique to reduce the function y(t) into a single sine term.

In polar form,

4sin(2πt + 0°) = 4 < 0°

In order to perform any mathematical operation, it is important to have both quantities in the same function either sine or cos

Convert the cos term into sine by adding phase shift of  90°

In polar form,

15cos(2πt + 90°) = 15sin(2πt) = 15 < 90°

Now add the two terms

y(t) = 4sin(2πt) + 15sin(2πt)

y(t) = 4 < 0°  + 15 < 90°

y(t) = 15.52 < 75.06°

y(t) = 15.52sin(2πt + 75.06°)

(b) Determine the amplitude, the period, the frequency in hertz of the function.

The standard form of a sinusoidal signal is given by

y(t) = Asin(2πft + Φ)

Where A is the amplitude, f is the frequency in hertz and ω = 2πf is angular frequency in rad/sec and Φ is the phase shift.

Comparing y(t) = 15.52sin(2πt + 75.06°) with the standard form yields,

Amplitude = A = 15.52

frequency = f = 1 Hz

Angular frequency = ω = 2π rad/s

Period = T = 1/f = 1/1 = 1 s

(c) Draw the function in time domain.

Please refer to the attached image. As you can see, the amplitude is around 15. frequency and period is 1 and phase shift is present too.