Given that , , and , where , find in the algebraic form.
First, use the fact that the graphs of and are horizontal translations of each other to write
To find the product, use the result that for two complex numbers
the product can be found using
Now use properties of the sine and cosine graphs again to write
Finally, use the fact that . Since , and , this means there is a right-angled triangle containing in which the opposite side is 3
units in length and the adjacent side is 4 units in length.
By Pythagoras' Theorem, the length of the hypotenuse is 5 units. Thus, and can be found using their definitions.
Finally, substitute these values into the expression for .