Use the form <math><mstyle displaystyle="true"><mi>a</mi><mi>sin</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.

Find the amplitude <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>a</mi><mo>|</mo></mrow></mstyle></math> .

Amplitude: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

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><mn>90</mn></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math>

Divide <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math>

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math>

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

Vertical Shift: <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math>

List the properties of the trigonometric function.

Amplitude: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

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

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math>

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mi>π</mi><mo>-</mo><mn>90</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mi>π</mi><mo>-</mo><mn>90</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> .

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

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

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

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mn>2</mn><mi>π</mi><mo>-</mo><mn>90</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi><mo>-</mo><mn>90</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract full rotations of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> until the angle is greater than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and less than <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

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

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Find the point at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>-</mo><mn>90</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Add <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract full rotations of <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> until the angle is greater than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and less than <math><mstyle displaystyle="true"><mn>2</mn><mi>π</mi></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

List the points in a table.

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

Amplitude: <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math>

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

Phase Shift: <math><mstyle displaystyle="true"><mo>-</mo><mn>90</mn></mstyle></math> (<math><mstyle displaystyle="true"><mn>90</mn></mstyle></math> to the left)

Vertical Shift: <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math>

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Name | eight hundred sixty-one million three hundred thirty-seven thousand six hundred forty-one |
---|

- 861337641 has 16 divisors, whose sum is
**1160928576** - The reverse of 861337641 is
**146733168** - Previous prime number is
**563**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 39
- Digital Root 3

Name | sixty-six million eight hundred fifty-five thousand two hundred sixty-two |
---|

- 66855262 has 8 divisors, whose sum is
**105561000** - The reverse of 66855262 is
**26255866** - Previous prime number is
**19**

- Is Prime? no
- Number parity even
- Number length 8
- Sum of Digits 40
- Digital Root 4

Name | one hundred forty-one million seventy thousand three hundred eighty-five |
---|

- 141070385 has 8 divisors, whose sum is
**176644800** - The reverse of 141070385 is
**583070141** - Previous prime number is
**5**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 29
- Digital Root 2