Trigonometry Examples

$- 3 \sin (π x - 2 x)$

Step 1

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = - 3$

$b = π - 2$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $- 3 \sin (π x - 2 x)$ .

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Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $π - 2$ in the formula for period.

$\frac{2 π}{| π - 2 |}$

Step 3.3

$π - 2$ is approximately $1.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π - 2}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{π - 2}$

Step 4.3

Divide $0$ by $π - 2$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $\frac{2 π}{π - 2}$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 \sin (π (0) - 2 \cdot 0)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Multiply $π$ by $0$ .

$f (0) = - 3 \sin (0 - 2 \cdot 0)$

Step 6.1.2.1.2

Multiply $- 2$ by $0$ .

$f (0) = - 3 \sin (0 + 0)$

Step 6.1.2.2

Add $0$ and $0$ .

$f (0) = - 3 \sin (0)$

Step 6.1.2.3

The exact value of $\sin (0)$ is $0$ .

$f (0) = - 3 \cdot 0$

Step 6.1.2.4

Multiply $- 3$ by $0$ .

$f (0) = 0$

Step 6.1.2.5

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = \frac{π}{2 (π - 2)}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{π}{2 (π - 2)}$ in the expression.

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (π (\frac{π}{2 (π - 2)}) - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $π (\frac{π}{2 (π - 2)})$ .

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Step 6.2.2.1.1.1

Combine $π$ and $\frac{π}{2 (π - 2)}$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π π}{2 (π - 2)} - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2.1.1.2

Raise $π$ to the power of $1$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π π}{2 (π - 2)} - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2.1.1.3

Raise $π$ to the power of $1$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π π}{2 (π - 2)} - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{1 + 1}}{2 (π - 2)} - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2.1.1.5

Add $1$ and $1$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - 2 \frac{π}{2 (π - 2)})$

Step 6.2.2.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.2.1

Factor $2$ out of $- 2$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} + 2 (- 1) (\frac{π}{2 (π - 2)}))$

Step 6.2.2.1.2.2

Cancel the common factor.

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} + 2 \cdot (- 1 \frac{π}{2 (π - 2)}))$

Step 6.2.2.1.2.3

Rewrite the expression.

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - 1 \frac{π}{π - 2})$

Step 6.2.2.1.3

Rewrite $- 1 \frac{π}{π - 2}$ as $- \frac{π}{π - 2}$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - \frac{π}{π - 2})$

Step 6.2.2.2

To write $- \frac{π}{π - 2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - \frac{π}{π - 2} \cdot \frac{2}{2})$

Step 6.2.2.3

Write each expression with a common denominator of $2 (π - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.3.1

Multiply $\frac{π}{π - 2}$ by $\frac{2}{2}$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - \frac{π \cdot 2}{(π - 2) \cdot 2})$

Step 6.2.2.3.2

Reorder the factors of $(π - 2) \cdot 2$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2}}{2 (π - 2)} - \frac{π \cdot 2}{2 (π - 2)})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2} - π \cdot 2}{2 (π - 2)})$

Step 6.2.2.5

Multiply $2$ by $- 1$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \sin (\frac{π^{2} - 2 π}{2 (π - 2)})$

Step 6.2.2.6

Evaluate $\sin (\frac{π^{2} - 2 π}{2 (π - 2)})$ .

$f (\frac{π}{2 (π - 2)}) = - 3 \cdot 1$

Step 6.2.2.7

Multiply $- 3$ by $1$ .

$f (\frac{π}{2 (π - 2)}) = - 3$

Step 6.2.2.8

The final answer is $- 3$ .

$- 3$

Step 6.3

Find the point at $x = \frac{π}{π - 2}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{π}{π - 2}$ in the expression.

$f (\frac{π}{π - 2}) = - 3 \sin (π (\frac{π}{π - 2}) - 2 \frac{π}{π - 2})$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Multiply $π (\frac{π}{π - 2})$ .

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Step 6.3.2.1.1.1

Combine $π$ and $\frac{π}{π - 2}$ .

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π π}{π - 2} - 2 \frac{π}{π - 2})$

Step 6.3.2.1.1.2

Raise $π$ to the power of $1$ .

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π π}{π - 2} - 2 \frac{π}{π - 2})$

Step 6.3.2.1.1.3

Raise $π$ to the power of $1$ .

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π π}{π - 2} - 2 \frac{π}{π - 2})$

Step 6.3.2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π^{1 + 1}}{π - 2} - 2 \frac{π}{π - 2})$

Step 6.3.2.1.1.5

Add $1$ and $1$ .

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π^{2}}{π - 2} - 2 \frac{π}{π - 2})$

Step 6.3.2.1.2

Combine $- 2$ and $\frac{π}{π - 2}$ .

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π^{2}}{π - 2} + \frac{- 2 π}{π - 2})$

Step 6.3.2.1.3

Move the negative in front of the fraction.

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π^{2}}{π - 2} - \frac{2 π}{π - 2})$

Step 6.3.2.2

Combine the numerators over the common denominator.

$f (\frac{π}{π - 2}) = - 3 \sin (\frac{π^{2} - 2 π}{π - 2})$

Step 6.3.2.3

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

$f (\frac{π}{π - 2}) = - 3 \sin (0)$

Step 6.3.2.4

The exact value of $\sin (0)$ is $0$ .

$f (\frac{π}{π - 2}) = - 3 \cdot 0$

Step 6.3.2.5

Multiply $- 3$ by $0$ .

$f (\frac{π}{π - 2}) = 0$

Step 6.3.2.6

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{3 π}{2 (π - 2)}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{3 π}{2 (π - 2)}$ in the expression.

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (π (\frac{3 π}{2 (π - 2)}) - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Multiply $π (\frac{3 π}{2 (π - 2)})$ .

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Step 6.4.2.1.1.1

Combine $π$ and $\frac{3 π}{2 (π - 2)}$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{π (3 π)}{2 (π - 2)} - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2.1.1.2

Raise $π$ to the power of $1$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 (π π)}{2 (π - 2)} - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2.1.1.3

Raise $π$ to the power of $1$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 (π π)}{2 (π - 2)} - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{1 + 1}}{2 (π - 2)} - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2.1.1.5

Add $1$ and $1$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - 2 \frac{3 π}{2 (π - 2)})$

Step 6.4.2.1.2

Cancel the common factor of $2$ .

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Step 6.4.2.1.2.1

Factor $2$ out of $- 2$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} + 2 (- 1) (\frac{3 π}{2 (π - 2)}))$

Step 6.4.2.1.2.2

Cancel the common factor.

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} + 2 \cdot (- 1 \frac{3 π}{2 (π - 2)}))$

Step 6.4.2.1.2.3

Rewrite the expression.

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - 1 \frac{3 π}{π - 2})$

Step 6.4.2.1.3

Rewrite $- 1 \frac{3 π}{π - 2}$ as $- \frac{3 π}{π - 2}$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - \frac{3 π}{π - 2})$

Step 6.4.2.2

To write $- \frac{3 π}{π - 2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - \frac{3 π}{π - 2} \cdot \frac{2}{2})$

Step 6.4.2.3

Write each expression with a common denominator of $2 (π - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.3.1

Multiply $\frac{3 π}{π - 2}$ by $\frac{2}{2}$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - \frac{3 π \cdot 2}{(π - 2) \cdot 2})$

Step 6.4.2.3.2

Reorder the factors of $(π - 2) \cdot 2$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2}}{2 (π - 2)} - \frac{3 π \cdot 2}{2 (π - 2)})$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2} - 3 π \cdot 2}{2 (π - 2)})$

Step 6.4.2.5

Multiply $2$ by $- 3$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \sin (\frac{3 π^{2} - 6 π}{2 (π - 2)})$

Step 6.4.2.6

Evaluate $\sin (\frac{3 π^{2} - 6 π}{2 (π - 2)})$ .

$f (\frac{3 π}{2 (π - 2)}) = - 3 \cdot - 1$

Step 6.4.2.7

Multiply $- 3$ by $- 1$ .

$f (\frac{3 π}{2 (π - 2)}) = 3$

Step 6.4.2.8

The final answer is $3$ .

$3$

Step 6.5

Find the point at $x = \frac{2 π}{π - 2}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{2 π}{π - 2}$ in the expression.

$f (\frac{2 π}{π - 2}) = - 3 \sin (π (\frac{2 π}{π - 2}) - 2 \frac{2 π}{π - 2})$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Multiply $π (\frac{2 π}{π - 2})$ .

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Step 6.5.2.1.1.1

Combine $π$ and $\frac{2 π}{π - 2}$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{π (2 π)}{π - 2} - 2 \frac{2 π}{π - 2})$

Step 6.5.2.1.1.2

Raise $π$ to the power of $1$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 (π π)}{π - 2} - 2 \frac{2 π}{π - 2})$

Step 6.5.2.1.1.3

Raise $π$ to the power of $1$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 (π π)}{π - 2} - 2 \frac{2 π}{π - 2})$

Step 6.5.2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{1 + 1}}{π - 2} - 2 \frac{2 π}{π - 2})$

Step 6.5.2.1.1.5

Add $1$ and $1$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{2}}{π - 2} - 2 \frac{2 π}{π - 2})$

Step 6.5.2.1.2

Multiply $- 2 \frac{2 π}{π - 2}$ .

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Step 6.5.2.1.2.1

Combine $- 2$ and $\frac{2 π}{π - 2}$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{2}}{π - 2} + \frac{- 2 (2 π)}{π - 2})$

Step 6.5.2.1.2.2

Multiply $2$ by $- 2$ .

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{2}}{π - 2} + \frac{- 4 π}{π - 2})$

Step 6.5.2.1.3

Move the negative in front of the fraction.

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{2}}{π - 2} - \frac{4 π}{π - 2})$

Step 6.5.2.2

Combine the numerators over the common denominator.

$f (\frac{2 π}{π - 2}) = - 3 \sin (\frac{2 π^{2} - 4 π}{π - 2})$

Step 6.5.2.3

Evaluate $\sin (\frac{2 π^{2} - 4 π}{π - 2})$ .

$f (\frac{2 π}{π - 2}) = - 3 \cdot 0$

Step 6.5.2.4

Multiply $- 3$ by $0$ .

$f (\frac{2 π}{π - 2}) = 0$

Step 6.5.2.5

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{2 (π - 2)} & - 3 \\ \frac{π}{π - 2} & 0 \\ \frac{3 π}{2 (π - 2)} & 3 \\ \frac{2 π}{π - 2} & 0 \end{array}$

Step 7

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

Amplitude: $3$

Period: $\frac{2 π}{π - 2}$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{2 (π - 2)} & - 3 \\ \frac{π}{π - 2} & 0 \\ \frac{3 π}{2 (π - 2)} & 3 \\ \frac{2 π}{π - 2} & 0 \end{array}$

Step 8