For any <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> , vertical asymptotes occur at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi></mstyle></math> , where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer. Use the basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>c</mi><mi>s</mi><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> , <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>)</mo></mrow></mstyle></math> , to find the vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> . Set the inside of the cosecant function, <math><mstyle displaystyle="true"><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> , for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>a</mi><mi>csc</mi><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the vertical asymptote occurs for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

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

Set the inside of the cosecant function <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> to the right side of the equation.

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

To write <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> from <math><mstyle displaystyle="true"><mn>6</mn><mi>π</mi></mstyle></math> .

The basic period for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> will occur at <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> , where <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> are vertical asymptotes.

Find the period <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> to find where the vertical asymptotes exist. Vertical asymptotes occur every half 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>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The vertical asymptotes for <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> occur at <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> , <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> , and every <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> , where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>csc</mi><mrow><mo>(</mo><mi>b</mi><mi>x</mi><mo>-</mo><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi></mstyle></math> to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Since the graph of the function <math><mstyle displaystyle="true"><mi>c</mi><mi>s</mi><mi>c</mi></mstyle></math> does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

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>1</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>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The phase shift of the function can be calculated from <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math>

Replace the values of <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> in the equation for phase shift.

Phase Shift: <math><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

Find the vertical shift <math><mstyle displaystyle="true"><mi>d</mi></mstyle></math> .

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: None

Period: <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> (<math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mi>π</mi><mi>n</mi></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is an integer

Amplitude: None

Period: <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> (<math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math>

Do you know how to Graph csc(x+pi/3)? 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 nine hundred twenty-one million fifty-nine thousand one hundred seventy-seven |
---|

- 1921059177 has 8 divisors, whose sum is
**2712083616** - The reverse of 1921059177 is
**7719501291** - Previous prime number is
**17**

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

Name | one billion five hundred fifty-four million sixty-six thousand nine hundred ninety-four |
---|

- 1554066994 has 4 divisors, whose sum is
**2331100494** - The reverse of 1554066994 is
**4996604551** - Previous prime number is
**2**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 49
- Digital Root 4

Name | nine hundred fifty-six million six hundred eighty-one thousand three hundred eighty-eight |
---|

- 956681388 has 128 divisors, whose sum is
**3615091200** - The reverse of 956681388 is
**883186659** - Previous prime number is
**13**

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