Let <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> . Substitute <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> for all occurrences of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>9</mn><mi>u</mi><mo>+</mo><mn>18</mn></mstyle></math> using the AC method.

Consider the form <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> . Find a pair of integers whose product is <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> . In this case, whose product is <math><mstyle displaystyle="true"><mn>18</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Write the factored form using these integers.

Replace all occurrences of <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> with <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

If any individual factor on the left side of the equation is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the entire expression will be equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> from both sides of the equation.

The range of cosine is <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>1</mn></mstyle></math> . Since <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> does not fall in this range, there is no solution.

No solution

No solution

No solution

The final solution is all the values that make <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mo>+</mo><mn>6</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> true.

No solution

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Name | one billion four hundred forty-seven million two hundred six thousand nine hundred eighty-nine |
---|

- 1447206989 has 16 divisors, whose sum is
**1581334272** - The reverse of 1447206989 is
**9896027441** - Previous prime number is
**293**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 50
- Digital Root 5

Name | two hundred ninety-four million nine hundred eighty thousand three hundred ninety-one |
---|

- 294980391 has 32 divisors, whose sum is
**545448960** - The reverse of 294980391 is
**193089492** - Previous prime number is
**79**

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

Name | one billion seven hundred twenty-nine million eight hundred forty thousand six hundred twenty-three |
---|

- 1729840623 has 8 divisors, whose sum is
**1841923776** - The reverse of 1729840623 is
**3260489271** - Previous prime number is
**61**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 42
- Digital Root 6