Question

A function f(x) is even if f(−x) = f(x). Which of sine or cosine is even? Explain how you know. (b) A function f(x) is odd if f(−x) = −f(x). Which of sine or cosine is odd? Explain how you know. (c) Cosine is a shift of sine. That is, cos(x) = sin(x + C) for some value of C. (Actually this is true for many many values of C.) Find such a value of C and explain how you know that you are correct.

Answer 1

a. Cosine is an even function

b. Sine is an odd function

c. cos(x) = sin(x + pi/2)

Explanation:

See the plot attached.

a. In the figure it can be seen that, for example, cos(pi/2) = cos(-pi/2); therefore, cosine is even.

b. In the figure it can be seen that, for example, sin(pi/2) = 1; sin(-pi/2) = -1; therefore, sine is odd.

c. Function displacements along x-axis are made adding (or subtracting) values to x; i. e., f(x + a), where a is a constant, displace the function f(x) a steps to the left. It can be seen in the figure that sine is cosine but displaced pi/2 values to the right. So, sin(x + pi/2) will displace sine pi/2 values to the left, where cosine is placed.