Convert to Trigonometric Form (5i)(2+6i)

Problem

(5*i)*(2+6*i)

Solution

1. Distribute the 5*i into the binomial (2+6*i)

5*i(2)+5*i*(6*i)

2. Simplify the terms using the property i2=−1

10*i+30*i2

10*i+30*(−1)

−30+10*i

3. Calculate the modulus r of the complex number z=a+b*i where a=−30 and b=10

r=√(,(−30)2+10)

r=√(,900+100)

r=√(,1000)

r=10√(,10)

4. Determine the argument θ using the formula tan(θ)=b/a

tan(θ)=10/(−30)

tan(θ)=−1/3

5. Identify the quadrant to find the correct value of θ Since a<0 and b>0 the point is in Quadrant II.

θ=arctan(−1/3)+π

θ≈2.82* radians

6. Write in trigonometric form using the structure z=r*(cos(θ)+i*sin(θ))

z=10√(,10)*(cos(2.82)+i*sin(2.82))

Final Answer

(5*i)*(2+6*i)=10√(,10)*(cos(2.82)+i*sin(2.82))

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