Trigonometry Examples

$y = \sin (\frac{1}{4} x + \frac{π}{3})$

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.

$a = 1$

$b = \frac{1}{4}$

$c = - \frac{π}{3}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\sin (\frac{x}{4} + \frac{π}{3})$ .

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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 $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 3.5

Multiply $4$ by $2$ .

$8 π$

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: $\frac{- \frac{π}{3}}{\frac{1}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{3} \cdot 4$

Step 4.4

Multiply $- \frac{π}{3} \cdot 4$ .

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Step 4.4.1

Multiply $4$ by $- 1$ .

Phase Shift: $- 4 \frac{π}{3}$

Step 4.4.2

Combine $- 4$ and $\frac{π}{3}$ .

Phase Shift: $\frac{- 4 π}{3}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{4 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $8 π$

Phase Shift: $- \frac{4 π}{3}$ ( $\frac{4 π}{3}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

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

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Step 6.1.1

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

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

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 the numerator by the reciprocal of the denominator.

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

Step 6.1.2.1.2

Cancel the common factor of $4$ .

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Step 6.1.2.1.2.1

Move the leading negative in $- \frac{4 π}{3}$ into the numerator.

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

Step 6.1.2.1.2.2

Factor $4$ out of $- 4 π$ .

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

Step 6.1.2.1.2.3

Cancel the common factor.

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

Step 6.1.2.1.2.4

Rewrite the expression.

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

Step 6.1.2.1.3

Move the negative in front of the fraction.

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

Step 6.1.2.2

Simplify terms.

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Step 6.1.2.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2.2

Add $- π$ and $π$ .

$f (- \frac{4 π}{3}) = \sin (\frac{0}{3})$

Step 6.1.2.2.3

Divide $0$ by $3$ .

$f (- \frac{4 π}{3}) = \sin (0)$

Step 6.1.2.3

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

$f (- \frac{4 π}{3}) = 0$

Step 6.1.2.4

The final answer is $0$ .

$0$

Step 6.2

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

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Step 6.2.1

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

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

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 the numerator by the reciprocal of the denominator.

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

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 $2 π$ .

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

Step 6.2.2.1.2.2

Factor $2$ out of $4$ .

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

Step 6.2.2.1.2.3

Cancel the common factor.

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

Step 6.2.2.1.2.4

Rewrite the expression.

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

Step 6.2.2.1.3

Multiply $\frac{π}{3}$ by $\frac{1}{2}$ .

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

Step 6.2.2.1.4

Multiply $3$ by $2$ .

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

Step 6.2.2.2

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

$f (\frac{2 π}{3}) = \sin (\frac{π}{6} + \frac{π}{3} \cdot \frac{2}{2})$

Step 6.2.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.3.1

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

$f (\frac{2 π}{3}) = \sin (\frac{π}{6} + \frac{π \cdot 2}{3 \cdot 2})$

Step 6.2.2.3.2

Multiply $3$ by $2$ .

$f (\frac{2 π}{3}) = \sin (\frac{π}{6} + \frac{π \cdot 2}{6})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (\frac{2 π}{3}) = \sin (\frac{π + π \cdot 2}{6})$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Move $2$ to the left of $π$ .

$f (\frac{2 π}{3}) = \sin (\frac{π + 2 \cdot π}{6})$

Step 6.2.2.5.2

Add $π$ and $2 π$ .

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

Step 6.2.2.6

Cancel the common factor of $3$ and $6$ .

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Step 6.2.2.6.1

Factor $3$ out of $3 π$ .

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

Step 6.2.2.6.2

Cancel the common factors.

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Step 6.2.2.6.2.1

Factor $3$ out of $6$ .

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

Step 6.2.2.6.2.2

Cancel the common factor.

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

Step 6.2.2.6.2.3

Rewrite the expression.

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

Step 6.2.2.7

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

$f (\frac{2 π}{3}) = 1$

Step 6.2.2.8

The final answer is $1$ .

$1$

Step 6.3

Find the point at $x = \frac{8 π}{3}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{8 π}{3}$ in the expression.

$f (\frac{8 π}{3}) = \sin (\frac{\frac{8 π}{3}}{4} + \frac{π}{3})$

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 the numerator by the reciprocal of the denominator.

$f (\frac{8 π}{3}) = \sin (\frac{8 π}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.3.2.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.1.2.1

Factor $4$ out of $8 π$ .

$f (\frac{8 π}{3}) = \sin (\frac{4 (2 π)}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.3.2.1.2.2

Cancel the common factor.

$f (\frac{8 π}{3}) = \sin (\frac{4 (2 π)}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.3.2.1.2.3

Rewrite the expression.

$f (\frac{8 π}{3}) = \sin (\frac{2 π}{3} + \frac{π}{3})$

Step 6.3.2.2

Simplify terms.

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Step 6.3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{8 π}{3}) = \sin (\frac{2 π + π}{3})$

Step 6.3.2.2.2

Add $2 π$ and $π$ .

$f (\frac{8 π}{3}) = \sin (\frac{3 π}{3})$

Step 6.3.2.2.3

Cancel the common factor of $3$ .

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Step 6.3.2.2.3.1

Cancel the common factor.

$f (\frac{8 π}{3}) = \sin (\frac{3 π}{3})$

Step 6.3.2.2.3.2

Divide $π$ by $1$ .

$f (\frac{8 π}{3}) = \sin (π)$

Step 6.3.2.3

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

$f (\frac{8 π}{3}) = \sin (0)$

Step 6.3.2.4

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

$f (\frac{8 π}{3}) = 0$

Step 6.3.2.5

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{14 π}{3}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{14 π}{3}$ in the expression.

$f (\frac{14 π}{3}) = \sin (\frac{\frac{14 π}{3}}{4} + \frac{π}{3})$

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 the numerator by the reciprocal of the denominator.

$f (\frac{14 π}{3}) = \sin (\frac{14 π}{3} \cdot \frac{1}{4} + \frac{π}{3})$

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 $14 π$ .

$f (\frac{14 π}{3}) = \sin (\frac{2 (7 π)}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.4.2.1.2.2

Factor $2$ out of $4$ .

$f (\frac{14 π}{3}) = \sin (\frac{2 (7 π)}{3} \cdot \frac{1}{2 (2)} + \frac{π}{3})$

Step 6.4.2.1.2.3

Cancel the common factor.

$f (\frac{14 π}{3}) = \sin (\frac{2 (7 π)}{3} \cdot \frac{1}{2 \cdot 2} + \frac{π}{3})$

Step 6.4.2.1.2.4

Rewrite the expression.

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{3} \cdot \frac{1}{2} + \frac{π}{3})$

Step 6.4.2.1.3

Multiply $\frac{7 π}{3}$ by $\frac{1}{2}$ .

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{3 \cdot 2} + \frac{π}{3})$

Step 6.4.2.1.4

Multiply $3$ by $2$ .

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{6} + \frac{π}{3})$

Step 6.4.2.2

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

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2})$

Step 6.4.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.3.1

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

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2})$

Step 6.4.2.3.2

Multiply $3$ by $2$ .

$f (\frac{14 π}{3}) = \sin (\frac{7 π}{6} + \frac{π \cdot 2}{6})$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (\frac{14 π}{3}) = \sin (\frac{7 π + π \cdot 2}{6})$

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Move $2$ to the left of $π$ .

$f (\frac{14 π}{3}) = \sin (\frac{7 π + 2 \cdot π}{6})$

Step 6.4.2.5.2

Add $7 π$ and $2 π$ .

$f (\frac{14 π}{3}) = \sin (\frac{9 π}{6})$

Step 6.4.2.6

Cancel the common factor of $9$ and $6$ .

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Step 6.4.2.6.1

Factor $3$ out of $9 π$ .

$f (\frac{14 π}{3}) = \sin (\frac{3 (3 π)}{6})$

Step 6.4.2.6.2

Cancel the common factors.

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Step 6.4.2.6.2.1

Factor $3$ out of $6$ .

$f (\frac{14 π}{3}) = \sin (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.6.2.2

Cancel the common factor.

$f (\frac{14 π}{3}) = \sin (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.6.2.3

Rewrite the expression.

$f (\frac{14 π}{3}) = \sin (\frac{3 π}{2})$

Step 6.4.2.7

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.

$f (\frac{14 π}{3}) = - \sin (\frac{π}{2})$

Step 6.4.2.8

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

$f (\frac{14 π}{3}) = - 1 \cdot 1$

Step 6.4.2.9

Multiply $- 1$ by $1$ .

$f (\frac{14 π}{3}) = - 1$

Step 6.4.2.10

The final answer is $- 1$ .

$- 1$

Step 6.5

Find the point at $x = \frac{20 π}{3}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{20 π}{3}$ in the expression.

$f (\frac{20 π}{3}) = \sin (\frac{\frac{20 π}{3}}{4} + \frac{π}{3})$

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 the numerator by the reciprocal of the denominator.

$f (\frac{20 π}{3}) = \sin (\frac{20 π}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.5.2.1.2

Cancel the common factor of $4$ .

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Step 6.5.2.1.2.1

Factor $4$ out of $20 π$ .

$f (\frac{20 π}{3}) = \sin (\frac{4 (5 π)}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.5.2.1.2.2

Cancel the common factor.

$f (\frac{20 π}{3}) = \sin (\frac{4 (5 π)}{3} \cdot \frac{1}{4} + \frac{π}{3})$

Step 6.5.2.1.2.3

Rewrite the expression.

$f (\frac{20 π}{3}) = \sin (\frac{5 π}{3} + \frac{π}{3})$

Step 6.5.2.2

Simplify terms.

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Step 6.5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{20 π}{3}) = \sin (\frac{5 π + π}{3})$

Step 6.5.2.2.2

Add $5 π$ and $π$ .

$f (\frac{20 π}{3}) = \sin (\frac{6 π}{3})$

Step 6.5.2.2.3

Cancel the common factor of $6$ and $3$ .

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Step 6.5.2.2.3.1

Factor $3$ out of $6 π$ .

$f (\frac{20 π}{3}) = \sin (\frac{3 (2 π)}{3})$

Step 6.5.2.2.3.2

Cancel the common factors.

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Step 6.5.2.2.3.2.1

Factor $3$ out of $3$ .

$f (\frac{20 π}{3}) = \sin (\frac{3 (2 π)}{3 (1)})$

Step 6.5.2.2.3.2.2

Cancel the common factor.

$f (\frac{20 π}{3}) = \sin (\frac{3 (2 π)}{3 \cdot 1})$

Step 6.5.2.2.3.2.3

Rewrite the expression.

$f (\frac{20 π}{3}) = \sin (\frac{2 π}{1})$

Step 6.5.2.2.3.2.4

Divide $2 π$ by $1$ .

$f (\frac{20 π}{3}) = \sin (2 π)$

Step 6.5.2.3

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

$f (\frac{20 π}{3}) = \sin (0)$

Step 6.5.2.4

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

$f (\frac{20 π}{3}) = 0$

Step 6.5.2.5

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{4 π}{3} & 0 \\ \frac{2 π}{3} & 1 \\ \frac{8 π}{3} & 0 \\ \frac{14 π}{3} & - 1 \\ \frac{20 π}{3} & 0 \end{array}$

Step 7

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

Amplitude: $1$

Period: $8 π$

Phase Shift: $- \frac{4 π}{3}$ ( $\frac{4 π}{3}$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{4 π}{3} & 0 \\ \frac{2 π}{3} & 1 \\ \frac{8 π}{3} & 0 \\ \frac{14 π}{3} & - 1 \\ \frac{20 π}{3} & 0 \end{array}$

Step 8