Question

Express (1+i)/ (√3+i) in the form x+iy, where x,y ∈ R. By writing each of 1+i and √3+i in polar form, deduce that cos(π/12) = (√3+1)/ 2√2 , sin(π/12) = (√3−1)/ 2√2 .

Answer 1

To express (1+i)/(√3+i) in the form x+iy, we can rationalize the denominator and simplify using polar form.

Step-by-step explanation:

To express (1+i)/(√3+i) in the form x+iy, we need to rationalize the denominator. First, express 1+i and √3+i in polar form. 1+i can be written as √2 * cis(45°) and √3+i can be written as 2 * cis(30°).

Now, we can substitute these values into the expression (1+i)/(√3+i) = (√2 * cis(45°))/(2 * cis(30°)).

Using the property of division in polar form (r1*cis(θ1))/(r2*cis(θ2)) = (r1/r2) * cis(θ1 - θ2), we can simplify the expression to (√2/2) * cis(45° - 30°).

Finally, simplifying the angle gives us (√2/2) * cis(15°).