Find the Cotangent Given the Point (( square root of 2)/3,( square root of 7)/3)

To find the <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> between the x-axis and the line between the points <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> , draw the triangle between the three points <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> , <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> , and <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Opposite : <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

Adjacent : <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

Multiply the numerator by the reciprocal of the denominator.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Combine <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> .

Combine and simplify the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>7</mn></msqrt></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><msqrt><mn>7</mn></msqrt></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> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><msqrt><mn>7</mn></msqrt></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>7</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>7</mn></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Evaluate the exponent.

Simplify the numerator.

Combine using the product rule for radicals.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

Approximate the result.

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Name | two billion eighty-one million twenty-two thousand two hundred ninety-six |
---|

- 2081022296 has 32 divisors, whose sum is
**7041992040** - The reverse of 2081022296 is
**6922201802** - Previous prime number is
**379**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 32
- Digital Root 5

Name | five hundred twenty-two million one hundred ninety-nine thousand three hundred twelve |
---|

- 522199312 has 64 divisors, whose sum is
**2673345060** - The reverse of 522199312 is
**213991225** - Previous prime number is
**89**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 34
- Digital Root 7

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

- 1315243617 has 4 divisors, whose sum is
**1753658160** - The reverse of 1315243617 is
**7163425131** - Previous prime number is
**3**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 33
- Digital Root 6