a trigonometric function.

Question

a) Give an example of an even trig function, proof that it is even. [2 marks]

b) Using your knowledge of transformations, transform your even trig function to the right to make it odd, then proof that it is odd. 

Answer 1

A) Of the two most basic trigonometric functions (sin, cos), Cosine is an even function. It is proven by the fact that cos(x) = cos(-x). In other words it doesn't matter if you move into the postive or negative direction, for every value of x the result will be the same.

For example:

Cos(60 deg) = 0.5 hint: 60 degrees is π/3 radians. π = 180 degrees

Cos(-60 deg) = 0.5

B) Sine is the odd function of the two basic trigonometric functions. It differs from Cosine because sin(x) != sin(-x). In reality, the sin(-x) = -sin(x). If you examine a graph of both functions closely, you will come to realize that they look identical, the only difference being that one is moved to the right by π/2 radians (90 degrees).

Since cosine is the even function, and we wish to move it to make it into the sine function we will have to use a phase shift. A phase shift simply changes whatever number you put into your function by a set amount, thus moving or shifting the function.

So in order to make Cosine into the Sine function, we simply need to decrease our input x by π/2 radians (or 90 degrees). Thus sin(x) = cos(x-π/2).

Now:

Cos(π/3 - π/2) = 0.86

Cos(-π/3 - π/2) = -0.86.

They are not equal in sign. The function is now odd.