comples number questions

[beezy4eva]

help with these questions would be appreciated please show working out for me

1. show that sinθ + cosθ i= cis (pi/2 -θ)

2. show that sinθ - cosθ i= cis (θ - pi/2)

3. simplify i(√3 -i)^7 giving the answer in modulus-Argument form.

[dcc]

Question 1:





Now we know that:

&

so we can express the above as:



Comparing to original, we see that they are equal!

Question 2:





We also know that:

&

Substituting into the above:



Which is the same as above!

Question 3:

Simplify:

Now, converting into polar form, we get:



and (using the inverse tangent function i.e. tan-1(-1/sqrt(3)))



so we can write z as such:



Now we wish to find:

so firstly:

using de Moivre's theorem:



(we must convert the angle as the principal argument of Z should be within )

converting this to cartesian co-ordinates, we get:





Now we are looking for i(z^7) so multiplying through by i:





Let b be i(z^7)

[beezy4eva]

thanks a tonne