Convert to Trigonometric Form 8i(7-5i)

Problem

8*i*(7−5*i)

Solution

1. Distribute the 8*i into the parentheses to simplify the expression into standard form a+b*i

8*i*(7−5*i)=56*i−40*i^2

2. Substitute i^2=−1 and simplify the real and imaginary parts.

56*i−40*(−1)=40+56*i

3. Calculate the modulus r using the formula r=√(,a^2+b^2) where a=40 and b=56

r=√(,40^2+56^2)

r=√(,1600+3136)

r=√(,4736)

r=8√(,74)

4. Find the argument θ using the formula tan(θ)=b/a Since the point (40,56) is in the first quadrant, θ=arctan(56/40)

θ=arctan(7/5)

5. Write the trigonometric form using the structure r*(cos(θ)+i*sin(θ))

8√(,74)*(cos(arctan(7/5))+i*sin(arctan(7/5)))

Final Answer

8*i*(7−5*i)=8√(,74)*(cos(arctan(7/5))+i*sin(arctan(7/5)))

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