Graph cos((2n+1)*90)

Graph cos((2n+1)*90)
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Find the amplitude .
Amplitude:
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 .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
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:
Cancel the common factor of and .
Factor out of .
Phase Shift:
Cancel the common factors.
Factor out of .
Phase Shift:
Cancel the common factor.
Phase Shift:
Rewrite the expression.
Phase Shift:
Phase Shift:
Phase Shift:
Move the negative in front of the fraction.
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
Select a few points to graph.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the distributive property.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the distributive property.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
The exact value of is .
Multiply by .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the distributive property.
Cancel the common factor of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the distributive property.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
List the points in a table.
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
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Name

Name one billion two hundred nineteen million six hundred twelve thousand four hundred thirty-two

Interesting facts

• 1219612432 has 64 divisors, whose sum is 6499252080
• The reverse of 1219612432 is 2342169121
• Previous prime number is 19

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 31
• Digital Root 4

Name

Name one billion thirty-seven million seven hundred forty-seven thousand two hundred twelve

Interesting facts

• 1037747212 has 32 divisors, whose sum is 2437975584
• The reverse of 1037747212 is 2127477301
• Previous prime number is 2441

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 34
• Digital Root 7

Name

Name one billion four hundred nine million eight hundred twenty thousand five hundred fifty-nine

Interesting facts

• 1409820559 has 8 divisors, whose sum is 1623523968
• The reverse of 1409820559 is 9550289041
• Previous prime number is 131

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 43
• Digital Root 7