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><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><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"><mo>-</mo><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.

Simplify terms.

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

Move the leading negative in <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> into the numerator.

Factor <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><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><mo>-</mo><mn>1</mn></mrow><mrow><msqrt><mn>7</mn></msqrt></mrow></mfrac></mstyle></math> .

Simplify the numerator.

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><msqrt><mn>2</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><msqrt><mn>2</mn></msqrt></mstyle></math> .

Move the negative in front of the fraction.

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 | six hundred fifty-two million two hundred ninety-four thousand eight hundred fifty-nine |
---|

- 652294859 has 8 divisors, whose sum is
**657388368** - The reverse of 652294859 is
**958492256** - Previous prime number is
**467**

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

Name | one billion eight hundred ninety million six hundred ninety-two thousand seven hundred fifty-seven |
---|

- 1890692757 has 8 divisors, whose sum is
**2007089856** - The reverse of 1890692757 is
**7572960981** - Previous prime number is
**17**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 54
- Digital Root 9

Name | one billion three hundred sixty-one million one hundred forty-seven thousand seven hundred fifty-five |
---|

- 1361147755 has 32 divisors, whose sum is
**1619896320** - The reverse of 1361147755 is
**5577411631** - Previous prime number is
**89**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 40
- Digital Root 4