This is the trigonometric form of a complex number where <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow></mstyle></math> is the modulus and <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> is the angle created on the complex plane.

The modulus of a complex number is the distance from the origin on the complex plane.

Substitute the actual values of <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>0</mn></mstyle></math> .

Raising <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to any positive power yields <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

Multiply by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Separate fractions.

Convert from <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> to <math><mstyle displaystyle="true"><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Simplify the expression.

Divide <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Apply the distributive property.

Simplify.

Multiply <math><mstyle displaystyle="true"><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> in terms of sines and cosines.

Apply the product rule to <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

One to any power is one.

Combine <math><mstyle displaystyle="true"><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

Convert from <math><mstyle displaystyle="true"><mfrac><mrow><msup><mi>cos</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow><mrow><msup><mi>sin</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> to <math><mstyle displaystyle="true"><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

Substitute the values of <math><mstyle displaystyle="true"><mi>θ</mi><mo>=</mo><mi>arctan</mi><mrow><mo>(</mo><mfrac><mrow><mn>0</mn></mrow><mrow><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mrow></mfrac><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>=</mo><msqrt><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>)</mo></mrow></msqrt></mstyle></math> .

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Name | seven hundred eighty-eight million five hundred five thousand eight hundred seventy-seven |
---|

- 788505877 has 4 divisors, whose sum is
**789963916** - The reverse of 788505877 is
**778505887** - Previous prime number is
**541**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 55
- Digital Root 1

Name | seven hundred twenty-eight million seven hundred seventy-two thousand eight hundred thirty-six |
---|

- 728772836 has 32 divisors, whose sum is
**1951426080** - The reverse of 728772836 is
**638277827** - Previous prime number is
**11**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 50
- Digital Root 5

Name | one billion three hundred forty-two million nine hundred forty-two thousand six hundred six |
---|

- 1342942606 has 4 divisors, whose sum is
**2014413912** - The reverse of 1342942606 is
**6062492431** - Previous prime number is
**2**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 37
- Digital Root 1