Question

The phase difference between sine and cosine function is 90 degrees

true or false

Phase shift of a sinusoid function can be either positive or negative.

true or false

Answer 1

NOTE: The sine function will be show using the red graph, and its step-by-step manipulations using the blue graph.

• A=amplitude

• /B=period

• C=phase shift or horizontal shift

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• D=vertical shift

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Let us suppose f₁(x) is a function formed by stretching the sine function vertically such that its amplitude becomes A, i.e., whose maximum distance from the x-axis is A. Then, f₁(x)=A sin(x).

Now, let f₂(x) be a function with vertical shift D from f₁(x), i.e., whose y coordinates for fixed x coordinates are D more than f₁(x). Then, for every value of x, f₂(x) exceeds f₁(x) by D,i.e., f₂(x)= f₁(x)+D=A sin(x)+D.

Now, let f₃(x) be a function with horizontal shift or phase shift C from f₂(x), i.e., whose x coordinates for fixed y coordinates are C more than f₂(x).i.e., f₃(x)=f₂(x+C). Therefore, f₃(x)= f₂(x+C)=A sin(x+C)+D.

Now, f₃(x) = A sin(x+C)+D = A sin(B(x/B)+C)+D. Let f₄(x) = f₃(x/B)

=> f₄(x) = A sin(B(x)+C)+D

Now, f₄(x+(/B)) = A sin(B(x+(/B))+C)+D = A sin(Bx+)+C)+D = A sin(Bx)+C)+D = f₄(x)

Therefore, the period of f₄(x) is /B.

​Hence, a function of the form has:

• A=amplitude
• /B=period
• C=phase shift
• D=vertical shift

from the sine function.

• The phase difference between sine and cosine function is 90 degrees: True or False?

Let us take and .

Then,

So, the phase shift of g(x) from sin(x) (=f(x)), or the phase difference between sin(x) and cos(x) is .

So, the answer is TRUE.

• Phase shift of a sinusoidal function can be either positive or negative: True or False?

Let us consider the function the sinusoidal function cos(x). Then, . So, cos(x) has positive phase shift.

Now, let us consider the sinusoidal function h(x)=-cos(x).

Then,

.

Therefore, h(x) has phase shift .

Then, phase shift of a sinusoidal function can be either positive or negative.

So, the answer is TRUE.