Trigonometry Examples

Find All Complex Number Solutions z = square root of 3-i
Step 1
This is the trigonometric form of a complex number where is the modulus and is the angle created on the complex plane.
Step 2
The modulus of a complex number is the distance from the origin on the complex plane.
where
Step 3
Substitute the actual values of and .
Step 4
Find .
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Step 4.1
Raise to the power of .
Step 4.2
Rewrite as .
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Step 4.2.1
Use to rewrite as .
Step 4.2.2
Apply the power rule and multiply exponents, .
Step 4.2.3
Combine and .
Step 4.2.4
Cancel the common factor of .
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Step 4.2.4.1
Cancel the common factor.
Step 4.2.4.2
Rewrite the expression.
Step 4.2.5
Evaluate the exponent.
Step 4.3
Simplify the expression.
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Step 4.3.1
Add and .
Step 4.3.2
Rewrite as .
Step 4.4
Pull terms out from under the radical, assuming positive real numbers.
Step 5
The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.
Step 6
Since inverse tangent of produces an angle in the fourth quadrant, the value of the angle is .
Step 7
Substitute the values of and .
Step 8
Replace the right side of the equation with the trigonometric form.