Convert to Trigonometric Form (8-7i)(6+7i)

Problem

(8−7*i)*(6+7*i)

Solution

1. Multiply the complex numbers using the FOIL method (First, Outer, Inner, Last).

(8)*(6)+(8)*(7*i)+(−7*i)*(6)+(−7*i)*(7*i)

2. Simplify the terms, remembering that i^2=−1

48+56*i−42*i−49*i^2

48+14*i−49*(−1)

48+14*i+49

97+14*i

3. Calculate the modulus r using the formula r=√(,a^2+b^2) where a=97 and b=14

r=√(,97^2+14^2)

r=√(,9409+196)

r=√(,9605)

4. Calculate the argument θ using the formula θ=arctan(b/a) Since the point (97,14) is in the first quadrant, no adjustment is needed.

θ=arctan(14/97)

θ≈8.21^∘

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

√(,9605)*(cos(arctan(14/97))+i*sin(arctan(14/97)))

Final Answer

(8−7*i)*(6+7*i)=√(,9605)*(cos(arctan(14/97))+i*sin(arctan(14/97)))

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