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

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)`