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Trigonometry Examples
Step 1
Use the Binomial Theorem.
Step 2
Step 2.1
Simplify each term.
Step 2.1.1
One to any power is one.
Step 2.1.2
One to any power is one.
Step 2.1.3
Multiply by .
Step 2.1.4
One to any power is one.
Step 2.1.5
Multiply by .
Step 2.1.6
Rewrite as .
Step 2.1.7
Multiply by .
Step 2.1.8
One to any power is one.
Step 2.1.9
Multiply by .
Step 2.1.10
Factor out .
Step 2.1.11
Rewrite as .
Step 2.1.12
Rewrite as .
Step 2.1.13
Multiply by .
Step 2.1.14
One to any power is one.
Step 2.1.15
Multiply by .
Step 2.1.16
Rewrite as .
Step 2.1.16.1
Rewrite as .
Step 2.1.16.2
Rewrite as .
Step 2.1.16.3
Raise to the power of .
Step 2.1.17
Multiply by .
Step 2.1.18
One to any power is one.
Step 2.1.19
Multiply by .
Step 2.1.20
Factor out .
Step 2.1.21
Rewrite as .
Step 2.1.21.1
Rewrite as .
Step 2.1.21.2
Rewrite as .
Step 2.1.21.3
Raise to the power of .
Step 2.1.22
Multiply by .
Step 2.1.23
One to any power is one.
Step 2.1.24
Multiply by .
Step 2.1.25
Factor out .
Step 2.1.26
Rewrite as .
Step 2.1.26.1
Rewrite as .
Step 2.1.26.2
Rewrite as .
Step 2.1.26.3
Raise to the power of .
Step 2.1.27
Multiply by .
Step 2.1.28
Rewrite as .
Step 2.1.29
Multiply by .
Step 2.1.30
Multiply by .
Step 2.1.31
Rewrite as .
Step 2.1.31.1
Factor out .
Step 2.1.31.2
Factor out .
Step 2.1.32
Rewrite as .
Step 2.1.32.1
Rewrite as .
Step 2.1.32.2
Rewrite as .
Step 2.1.32.3
Raise to the power of .
Step 2.1.33
Multiply by .
Step 2.1.34
Rewrite as .
Step 2.1.35
Rewrite as .
Step 2.1.36
Multiply by .
Step 2.1.37
Rewrite as .
Step 2.1.38
Rewrite as .
Step 2.1.38.1
Rewrite as .
Step 2.1.38.2
Rewrite as .
Step 2.1.38.3
Raise to the power of .
Step 2.1.39
One to any power is one.
Step 2.2
Simplify by adding terms.
Step 2.2.1
Subtract from .
Step 2.2.2
Simplify by adding and subtracting.
Step 2.2.2.1
Add and .
Step 2.2.2.2
Subtract from .
Step 2.2.2.3
Add and .
Step 2.2.3
Subtract from .
Step 2.2.4
Add and .
Step 2.2.5
Subtract from .
Step 2.2.6
Add and .
Step 3
This is the trigonometric form of a complex number where is the modulus and is the angle created on the complex plane.
Step 4
The modulus of a complex number is the distance from the origin on the complex plane.
where
Step 5
Substitute the actual values of and .
Step 6
Step 6.1
Raising to any positive power yields .
Step 6.2
Raise to the power of .
Step 6.3
Add and .
Step 6.4
Rewrite as .
Step 6.5
Pull terms out from under the radical, assuming positive real numbers.
Step 7
The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.
Step 8
Since inverse tangent of produces an angle in the first quadrant, the value of the angle is .
Step 9
Substitute the values of and .