Convert to Trigonometric Form i^167

Problem

i167

Solution

1. Simplify the power of i by using the property that i4=1

2. Divide the exponent 167 by 4 to find the remainder.

167=4×41+3

3. Apply the remainder to the base i

i167=(i4)41×i3

i167=1×(−i)

i167=−i

4. Identify the complex number in the form a+b*i

z=0−1*i

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

r=√(,0+(−1)2)

r=1

6. Determine the argument θ by locating the point (0,−1) on the complex plane.

θ=(3*π)/2

7. Write the result in trigonometric form r*(cos(θ)+i*sin(θ))

1*(cos((3*π)/2)+i*sin((3*π)/2))

Final Answer

i167=cos((3*π)/2)+i*sin((3*π)/2)

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