Trigonometry Examples

$y = 1 + 3 \sin (2 x - \frac{π}{2})$

Step 1

Rewrite the expression as $3 \sin (2 x - \frac{π}{2}) + 1$ .

$3 \sin (2 x - \frac{π}{2}) + 1$

Step 2

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

$a = 3$

$b = 2$

$c = \frac{π}{2}$

$d = 1$

Step 3

Find the amplitude $| a |$ .

Amplitude: $3$

Step 4

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 4.1

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

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Step 4.1.1

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

$\frac{2 π}{| b |}$

Step 4.1.2

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

$\frac{2 π}{| 2 |}$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.1.4.2

Divide $π$ by $1$ .

$π$

Step 4.2

Find the period of $1$ .

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Step 4.2.1

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

$\frac{2 π}{| b |}$

Step 4.2.2

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

$\frac{2 π}{| 2 |}$

Step 4.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.2.4.2

Divide $π$ by $1$ .

$π$

Step 4.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$π$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

Phase Shift: $\frac{\frac{π}{2}}{2}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{2}$

Step 5.4

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 5.4.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

Phase Shift: $\frac{π}{2 \cdot 2}$

Step 5.4.2

Multiply $2$ by $2$ .

Phase Shift: $\frac{π}{4}$

Step 6

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $1$

Step 7

Select a few points to graph.

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Step 7.1

Find the point at $x = \frac{π}{4}$ .

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Step 7.1.1

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

$f (\frac{π}{4}) = 1 + 3 \sin (2 (\frac{π}{4}) - \frac{π}{2})$

Step 7.1.2

Simplify the result.

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Step 7.1.2.1

Simplify each term.

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Step 7.1.2.1.1

Cancel the common factor of $2$ .

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Step 7.1.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{π}{4}) = 1 + 3 \sin (2 (\frac{π}{2 (2)}) - \frac{π}{2})$

Step 7.1.2.1.1.2

Cancel the common factor.

$f (\frac{π}{4}) = 1 + 3 \sin (2 (\frac{π}{2 \cdot 2}) - \frac{π}{2})$

Step 7.1.2.1.1.3

Rewrite the expression.

$f (\frac{π}{4}) = 1 + 3 \sin (\frac{π}{2} - \frac{π}{2})$

Step 7.1.2.1.2

Combine the numerators over the common denominator.

$f (\frac{π}{4}) = 1 + 3 \sin (\frac{π - π}{2})$

Step 7.1.2.1.3

Subtract $π$ from $π$ .

$f (\frac{π}{4}) = 1 + 3 \sin (\frac{0}{2})$

Step 7.1.2.1.4

Divide $0$ by $2$ .

$f (\frac{π}{4}) = 1 + 3 \sin (0)$

Step 7.1.2.1.5

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

$f (\frac{π}{4}) = 1 + 3 \cdot 0$

Step 7.1.2.1.6

Multiply $3$ by $0$ .

$f (\frac{π}{4}) = 1 + 0$

Step 7.1.2.2

Add $1$ and $0$ .

$f (\frac{π}{4}) = 1$

Step 7.1.2.3

The final answer is $1$ .

$1$

Step 7.2

Find the point at $x = \frac{π}{2}$ .

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Step 7.2.1

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

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

Step 7.2.2

Simplify the result.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1.1

Cancel the common factor.

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

Step 7.2.2.1.1.2

Rewrite the expression.

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

Step 7.2.2.1.2

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

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

Step 7.2.2.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 7.2.2.1.4

Combine the numerators over the common denominator.

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

Step 7.2.2.1.5

Simplify the numerator.

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Step 7.2.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 7.2.2.1.5.2

Subtract $π$ from $2 π$ .

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

Step 7.2.2.1.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{π}{2}) = 1 + 3 \cdot 1$

Step 7.2.2.1.7

Multiply $3$ by $1$ .

$f (\frac{π}{2}) = 1 + 3$

Step 7.2.2.2

Add $1$ and $3$ .

$f (\frac{π}{2}) = 4$

Step 7.2.2.3

The final answer is $4$ .

$4$

Step 7.3

Find the point at $x = \frac{3 π}{4}$ .

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Step 7.3.1

Replace the variable $x$ with $\frac{3 π}{4}$ in the expression.

$f (\frac{3 π}{4}) = 1 + 3 \sin (2 (\frac{3 π}{4}) - \frac{π}{2})$

Step 7.3.2

Simplify the result.

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Step 7.3.2.1

Simplify each term.

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Step 7.3.2.1.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{3 π}{4}) = 1 + 3 \sin (2 (\frac{3 π}{2 (2)}) - \frac{π}{2})$

Step 7.3.2.1.1.2

Cancel the common factor.

$f (\frac{3 π}{4}) = 1 + 3 \sin (2 (\frac{3 π}{2 \cdot 2}) - \frac{π}{2})$

Step 7.3.2.1.1.3

Rewrite the expression.

$f (\frac{3 π}{4}) = 1 + 3 \sin (\frac{3 π}{2} - \frac{π}{2})$

Step 7.3.2.1.2

Combine the numerators over the common denominator.

$f (\frac{3 π}{4}) = 1 + 3 \sin (\frac{3 π - π}{2})$

Step 7.3.2.1.3

Subtract $π$ from $3 π$ .

$f (\frac{3 π}{4}) = 1 + 3 \sin (\frac{2 π}{2})$

Step 7.3.2.1.4

Cancel the common factor of $2$ .

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Step 7.3.2.1.4.1

Cancel the common factor.

$f (\frac{3 π}{4}) = 1 + 3 \sin (\frac{2 π}{2})$

Step 7.3.2.1.4.2

Divide $π$ by $1$ .

$f (\frac{3 π}{4}) = 1 + 3 \sin (π)$

Step 7.3.2.1.5

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

$f (\frac{3 π}{4}) = 1 + 3 \sin (0)$

Step 7.3.2.1.6

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

$f (\frac{3 π}{4}) = 1 + 3 \cdot 0$

Step 7.3.2.1.7

Multiply $3$ by $0$ .

$f (\frac{3 π}{4}) = 1 + 0$

Step 7.3.2.2

Add $1$ and $0$ .

$f (\frac{3 π}{4}) = 1$

Step 7.3.2.3

The final answer is $1$ .

$1$

Step 7.4

Find the point at $x = π$ .

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Step 7.4.1

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

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

Step 7.4.2

Simplify the result.

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Step 7.4.2.1

Simplify each term.

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Step 7.4.2.1.1

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

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

Step 7.4.2.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 7.4.2.1.3

Combine the numerators over the common denominator.

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

Step 7.4.2.1.4

Simplify the numerator.

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Step 7.4.2.1.4.1

Multiply $2$ by $2$ .

$f (π) = 1 + 3 \sin (\frac{4 π - π}{2})$

Step 7.4.2.1.4.2

Subtract $π$ from $4 π$ .

$f (π) = 1 + 3 \sin (\frac{3 π}{2})$

Step 7.4.2.1.5

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.

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

Step 7.4.2.1.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (π) = 1 + 3 (- 1 \cdot 1)$

Step 7.4.2.1.7

Multiply $3 (- 1 \cdot 1)$ .

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Step 7.4.2.1.7.1

Multiply $- 1$ by $1$ .

$f (π) = 1 + 3 \cdot - 1$

Step 7.4.2.1.7.2

Multiply $3$ by $- 1$ .

$f (π) = 1 - 3$

Step 7.4.2.2

Subtract $3$ from $1$ .

$f (π) = - 2$

Step 7.4.2.3

The final answer is $- 2$ .

$- 2$

Step 7.5

Find the point at $x = \frac{5 π}{4}$ .

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Step 7.5.1

Replace the variable $x$ with $\frac{5 π}{4}$ in the expression.

$f (\frac{5 π}{4}) = 1 + 3 \sin (2 (\frac{5 π}{4}) - \frac{π}{2})$

Step 7.5.2

Simplify the result.

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Step 7.5.2.1

Simplify each term.

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Step 7.5.2.1.1

Cancel the common factor of $2$ .

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Step 7.5.2.1.1.1

Factor $2$ out of $4$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (2 (\frac{5 π}{2 (2)}) - \frac{π}{2})$

Step 7.5.2.1.1.2

Cancel the common factor.

$f (\frac{5 π}{4}) = 1 + 3 \sin (2 (\frac{5 π}{2 \cdot 2}) - \frac{π}{2})$

Step 7.5.2.1.1.3

Rewrite the expression.

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{5 π}{2} - \frac{π}{2})$

Step 7.5.2.1.2

Combine the numerators over the common denominator.

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{5 π - π}{2})$

Step 7.5.2.1.3

Subtract $π$ from $5 π$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{4 π}{2})$

Step 7.5.2.1.4

Cancel the common factor of $4$ and $2$ .

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Step 7.5.2.1.4.1

Factor $2$ out of $4 π$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{2 (2 π)}{2})$

Step 7.5.2.1.4.2

Cancel the common factors.

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Step 7.5.2.1.4.2.1

Factor $2$ out of $2$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{2 (2 π)}{2 (1)})$

Step 7.5.2.1.4.2.2

Cancel the common factor.

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{2 (2 π)}{2 \cdot 1})$

Step 7.5.2.1.4.2.3

Rewrite the expression.

$f (\frac{5 π}{4}) = 1 + 3 \sin (\frac{2 π}{1})$

Step 7.5.2.1.4.2.4

Divide $2 π$ by $1$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (2 π)$

Step 7.5.2.1.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{5 π}{4}) = 1 + 3 \sin (0)$

Step 7.5.2.1.6

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

$f (\frac{5 π}{4}) = 1 + 3 \cdot 0$

Step 7.5.2.1.7

Multiply $3$ by $0$ .

$f (\frac{5 π}{4}) = 1 + 0$

Step 7.5.2.2

Add $1$ and $0$ .

$f (\frac{5 π}{4}) = 1$

Step 7.5.2.3

The final answer is $1$ .

$1$

Step 7.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{4} & 1 \\ \frac{π}{2} & 4 \\ \frac{3 π}{4} & 1 \\ π & - 2 \\ \frac{5 π}{4} & 1 \end{array}$

Step 8

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

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $1$

$\begin{array}{c} x & f (x) \\ \frac{π}{4} & 1 \\ \frac{π}{2} & 4 \\ \frac{3 π}{4} & 1 \\ π & - 2 \\ \frac{5 π}{4} & 1 \end{array}$

Step 9