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

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

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

Raise <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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>y</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> .

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

Rewrite <math><mstyle displaystyle="true"><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><msup><mi>csc</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> in a factored form.

Rewrite the middle term.

Rearrange terms.

Factor first three terms by perfect square rule.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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"><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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> .

Multiply <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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> .

Reorder the factors of <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

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

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> , where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Pull terms out from under the radical, assuming positive real numbers.

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>y</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>2</mn><mi>y</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><mi>csc</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>cot</mi><mrow><mo>(</mo><mn>2</mn><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

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Name | one billion seven hundred seventy-five million six thousand six hundred eighty-five |
---|

- 1775006685 has 16 divisors, whose sum is
**2388394800** - The reverse of 1775006685 is
**5866005771** - Previous prime number is
**109**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 45
- Digital Root 9

Name | six hundred thirty million forty-five thousand two hundred fifty |
---|

- 630045250 has 32 divisors, whose sum is
**1211339376** - The reverse of 630045250 is
**052540036** - Previous prime number is
**25**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 25
- Digital Root 7

Name | eight hundred nine million one hundred twenty-seven thousand seven hundred sixteen |
---|

- 809127716 has 32 divisors, whose sum is
**1929733632** - The reverse of 809127716 is
**617721908** - Previous prime number is
**991**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 41
- Digital Root 5