Graph 4/5*csc(3/2x-pi/6)

Graph 4/5*csc(3/2x-pi/6)
Find the asymptotes.
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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 .
Solve for .
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Add to both sides of the equation.
Multiply both sides of the equation by .
Simplify both sides of the equation.
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Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply and .
Multiply by .
Set the inside of the cosecant function equal to .
Solve for .
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Move all terms not containing to the right side of the equation.
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Add to 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.
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Multiply by .
Add and .
Multiply both sides of the equation by .
Simplify both sides of the equation.
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Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply and .
Multiply by .
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.
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is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Combine and .
Multiply by .
Combine and .
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 .
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The period of the function can be calculated using .
Replace with in the formula for period.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Combine and .
Multiply by .
Combine and .
Find the phase shift using the formula .
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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:
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Cancel the common factor of .
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Factor out of .
Phase Shift:
Cancel the common factor.
Phase Shift:
Rewrite the expression.
Phase Shift:
Phase Shift:
Multiply and .
Phase Shift:
Multiply 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 right)
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 right)
Vertical Shift:
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Name

Name eight hundred ninety-five million nine hundred twenty-eight thousand nine hundred forty-five

Interesting facts

  • 895928945 has 16 divisors, whose sum is 1132847040
  • The reverse of 895928945 is 549829598
  • Previous prime number is 1117

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 59
  • Digital Root 5

Name

Name five hundred twenty-six million three hundred fourteen thousand eight hundred seventy-one

Interesting facts

  • 526314871 has 4 divisors, whose sum is 526368744
  • The reverse of 526314871 is 178413625
  • Previous prime number is 12821

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 37
  • Digital Root 1

Name

Name sixty-four million nine hundred forty-one thousand five hundred forty-three

Interesting facts

  • 64941543 has 16 divisors, whose sum is 115662976
  • The reverse of 64941543 is 34514946
  • Previous prime number is 571

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 8
  • Sum of Digits 36
  • Digital Root 9