62 C H A P T E R 0: From the Ground Up!
(c) Show that the hyperbolic cosine is always positive and bigger than 1 for all values ofθ.
(d) Show thatsinh(θ)=−sinh(−θ).
(e) Write a MATLAB script to compute and plot these functions between−10 and 10.
0.23. Phasors!
A phasor can be thought of as a vector, representing a complex number, rotating around the polar plane
at a certain frequency expressed in radians/sec. The projection of such a vector onto the real axis gives a
cosine. This problem will show the algebra of phasors, which would help you with some of the trigonometric
identities that are hard to remember.
(a) When you plot a sine signaly(t)=Asin( 0 t), you notice that it is a cosinex(t)=Acos( 0 t)shifted in
time—that is,y(t)=Asin( 0 t)=Acos( 0 (t− (^1) t))=x(t− (^1) t)
How much is this shift (^1) t? Better yet, what is (^1) θ= 01 tor the shift in phase? One thus only need to
consider cosine functions with different phase shifts instead of sines and cosines.
(b) You should have found the answer above is (^1) θ=π/ 2 (if not, go back and try it and see if it works).
Thus, the phasor that generatesx(t)=Acos( 0 t)isAej^0 so thatx(t)=Re[Aej^0 ej^0 t]. The phasor
corresponding to the siney(t)should then beAe−jπ/^2. Obtain an expression fory(t)similar to the one
forx(t)in terms of this phasor.
(c) According to the above results, give the phasors corresponding to−x(t)=−Acos( 0 t)and−y(t)=
−sin( 0 t). Plot the phasors that generatecos, sin,−cos, and−sinfor a given frequency. Do you see
now how these functions are connected? How many radians do you need to shift in a positive or
negative direction to get a sine from a cosine, etc.
(d) Suppose then you have the sum of two sinusoids, for instancez(t)=x(t)+y(t), adding the corre-
sponding phasors forx(t)andy(t)at some time (e.g.,t= 0 ), which is just a sum of two vectors, you
should get a vector and the corresponding phasor. Get the phasor forz(t)and the expression for it in
terms of a cosine.