Find Where Undefined/Discontinuous (1+cos(3t))/(sin(3t))+(sin(3t))/(1+cos(3t))

Set the denominator in <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is undefined.

Take the inverse sine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>t</mi></mstyle></math> from inside the sine.

The exact value of <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>t</mi><mo>=</mo><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

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

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>t</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> to find the solution in the second quadrant.

Simplify the expression to find the second solution.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>t</mi><mo>=</mo><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

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

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>t</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Find the period.

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> in the formula for period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

The period of the <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> radians in both directions.

Consolidate the answers.

Set the denominator in <math><mstyle displaystyle="true"><mfrac><mrow><mi>sin</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is undefined.

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from both sides of the equation.

Take the inverse cosine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>t</mi></mstyle></math> from inside the cosine.

The exact value of <math><mstyle displaystyle="true"><mi>arccos</mi><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>t</mi><mo>=</mo><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

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

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>t</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> to find the solution in the third quadrant.

Simplify the expression to find the second solution.

Subtract <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> from <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Divide each term by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>t</mi><mo>=</mo><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

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

Cancel the common factor.

Find the period.

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> in the formula for period.

The absolute value is the distance between a number and zero. The distance between <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

The period of the <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> radians in both directions.

The equation is undefined where the denominator equals <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the argument of a square root is less than <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , or the argument of a logarithm is less than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

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Do you know how to Find Where Undefined/Discontinuous (1+cos(3t))/(sin(3t))+(sin(3t))/(1+cos(3t))? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one billion four hundred eleven million nine hundred forty-two thousand four hundred seventy-two |
---|

- 1411942472 has 32 divisors, whose sum is
**4855218732** - The reverse of 1411942472 is
**2742491141** - Previous prime number is
**53**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 35
- Digital Root 8

Name | one billion two hundred seventy-four million one hundred thousand nine hundred twenty-one |
---|

- 1274100921 has 16 divisors, whose sum is
**1924569920** - The reverse of 1274100921 is
**1290014721** - Previous prime number is
**51**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 27
- Digital Root 9

Name | one billion nine hundred sixty million eight hundred ten thousand seven hundred thirty-six |
---|

- 1960810736 has 128 divisors, whose sum is
**10519773264** - The reverse of 1960810736 is
**6370180691** - Previous prime number is
**4861**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 41
- Digital Root 5