# Trigonometry Study Guide? Help please?

3.
find the quotient z1/z2 of the complex numbers. Leave answer in polar form.
z1=5(cos 200 = isin 200)
z2=4(cos50=isin50)

4.
Use DeMoivre's Theorem to find the indicated power of the complex number. Write answer in rectangular form.

[3(cos 15+isin15)]^4

3)Given that,

z1 = 5(cos(200º) + i*sin(200º))

= 5(cos(10π/9) + i*sin(10π/9))

= 5e^(10πi/9)                                [where acosѲ + i*sinѲ = e^Ѳ]

z2 = 4(cos(50º) + i*sin(50º))

= 4(cos(5π/18) + i*sin(5π/18))

= 4e^(5πi/18)                                [where acosѲ + i*sinѲ = e^Ѳ]

Therefore z1/z2 = (5e^(10πi/9))/(4e^(5πi/18))

= (5/4)e^(10πi/9 - 5πi/18)

= (5/4)e^((20πi - 5πi)/18)

= (5/4)e^(15πi/18)

= (5/4)e^(5πi/6)

= (5/4)(cos(5π/6) + i*sin(5π/6))

= (5/4)(cos(150º) + i*sin(150º)).

4) The De Moivre's theorem states that, if n be any real number, then (cosθ+isinθ)n=cosnθ+isinnθ.
Now we have to find the indicated power of (3(cos(15º) + i*sin(15º)))^4

= (3^4)(cos(4*15º) + i*sin(4*15º))

= 81(cos(60º) + i*sin(60º))

= 81((√3)/2 + i/2)                        [ Where cos(60º)=√3)/2 and sin(60º)=1/2 ]

= (81√3)/2 + i*(81/2)

Therefore the answer is (81√3)/2 + i*(81/2).