Convert to Trigonometric Form 2+7i+(-6+i)

Problem

2+7*i+(−6+i)

Solution

1. Simplify the complex expression by combining the real parts and the imaginary parts.

2−6=−4

7*i+i=8*i

z=−4+8*i

2. Calculate the modulus r using the formula r=√(,a2+b2) where a=−4 and b=8

r=√(,(−4)2+8)

r=√(,16+64)

r=√(,80)

r=4√(,5)

3. Determine the argument θ using the formula tan(θ)=b/a Since the point (−4,8) is in the second quadrant, we add π (or 180 to the reference angle.

tan(θ)=8/(−4)

tan(θ)=−2

θ=arctan(−2)+π

θ≈2.034* radians

4. Express the complex number in trigonometric form z=r*(cos(θ)+i*sin(θ))

z=4√(,5)*(cos(2.034)+i*sin(2.034))

Final Answer

2+7*i+(−6+i)=4√(,5)*(cos(2.034)+i*sin(2.034))

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