Question

Explain why you cannot use the Distributive Property to evaluate an expression in the form of sin(a + b). Please illustrate with an example along with your explanation.

Answer 1

Answer:

The distributive property is to multiply the outside of a parenthesis by each term inside, for example:

$x(y+z)=xy+xz$

this cannot be done with the trigonometric functions such as the sine.

An example of this:

let's prove that we don't get the correct result using the distributive property in the following expression:

$sin (\pi+\pi/2)$ ≠ $sin(\pi)+sin(\pi/2)$

We add the elements in the parentheses on the left side:

$sin(3\pi/2)$  ≠ $sin(\pi)+sin(\pi/2)$

this are known values of the sine function:

$sin(3\pi/2)=-1$

$sin(\pi)=0$

$sin(\pi/2)=1$

substituting these values we will get that:

-1  ≠ 0 +1

-1  ≠ 1

Thus we notice that we don't get the correct result using the distributive property.

The correct way to express the angle sum in the sine function is:

$sin(a+b)=sin(a)cos(b)+sin(b)cos(a)$