Convert to Trigonometric Form (6/(6+i))((6-i)/(6-i))

Problem

(6/(6+i))*((6−i)/(6−i))

Solution

1. Multiply the numerators and denominators to simplify the expression into a single fraction.

(6*(6−i))/((6+i)*(6−i))

2. Distribute the 6 in the numerator and apply the difference of squares formula (a+b*i)*(a−b*i)=a2+b2 to the denominator.

(36−6*i)/(6+1)

3. Simplify the denominator and write the complex number in standard form a+b*i

(36−6*i)/37

36/37−6/37*i

4. Calculate the modulus r using the formula r=√(,a2+b2)

r=√(,(36/37)2+(−6/37)2)

r=√(,1296/1369+36/1369)

r=√(,1332/1369)

r=√(,36⋅37)/37

r=(6√(,37))/37

5. Determine the argument θ using tan(θ)=b/a Since a>0 and b<0 the angle is in the fourth quadrant.

tan(θ)=(−6/37)/(36/37)

tan(θ)=−1/6

θ=arctan(−1/6)

6. Express the result in trigonometric form r*(cos(θ)+i*sin(θ))

(6√(,37))/37*(cos(arctan(−1/6))+i*sin(arctan(−1/6)))

Final Answer

(6/(6+i))*((6−i)/(6−i))=(6√(,37))/37*(cos(arctan(−1/6))+i*sin(arctan(−1/6)))

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