# Graph csc(x+pi/3)

Graph csc(x+pi/3)
Find the asymptotes.
For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the cosecant function, , for equal to to find where the vertical asymptote occurs for .
Subtract from both sides of the equation.
Set the inside of the cosecant function equal to .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
The basic period for will occur at , where and are vertical asymptotes.
Find the period 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 and is .
Divide by .
The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.
Cosecant only has vertical asymptotes.
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
Find the period of .
The period of the function can be calculated using .
Replace with in the formula for period.
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
Find the phase shift using the formula .
The phase shift of the function can be calculated from .
Phase Shift:
Replace the values of and in the equation for phase shift.
Phase Shift:
Divide by .
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: ( to the left)
Vertical Shift:
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: where is an integer
Amplitude: None
Period:
Phase Shift: ( to the left)
Vertical Shift:
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### Name

Name one billion nine hundred twenty-one million fifty-nine thousand one hundred seventy-seven

### Interesting facts

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

### Basic properties

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

### Name

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

### Interesting facts

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

### Basic properties

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

### Name

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

### Interesting facts

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

### Basic properties

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