Trigonometry Examples

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

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.

$a = 3$

$b = \frac{π}{4}$

$c = - \frac{3 π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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

$\frac{2 π}{| \frac{π}{4} |}$

Step 3.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 3.5

Cancel the common factor of $π$ .

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Step 3.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 3.5.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 3.5.3

Rewrite the expression.

$2 \cdot 4$

Step 3.6

Multiply $2$ by $4$ .

$8$

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: $\frac{- \frac{3 π}{4}}{\frac{π}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{3 π}{4} \cdot \frac{4}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

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

Phase Shift: $\frac{- 3 π}{4} \cdot \frac{4}{π}$

Step 4.4.2

Factor $π$ out of $- 3 π$ .

Phase Shift: $\frac{π \cdot - 3}{4} \cdot \frac{4}{π}$

Step 4.4.3

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 3}{4} \cdot \frac{4}{π}$

Step 4.4.4

Rewrite the expression.

Phase Shift: $\frac{- 3}{4} \cdot 4$

Step 4.5

Cancel the common factor of $4$ .

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Step 4.5.1

Cancel the common factor.

Phase Shift: $\frac{- 3}{4} \cdot 4$

Step 4.5.2

Rewrite the expression.

Phase Shift: $- 3$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $8$

Phase Shift: $- 3$ ( $3$ 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 $x = - 3$ .

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Step 6.1.1

Replace the variable $x$ with $- 3$ in the expression.

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

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2

Move $- 3$ to the left of $π$ .

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

Step 6.1.2.3

Simplify by adding terms.

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Step 6.1.2.3.1

Add $- 3 π$ and $3 π$ .

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

Step 6.1.2.3.2

Divide $0$ by $4$ .

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

Step 6.1.2.4

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

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

Step 6.1.2.5

Multiply $3$ by $0$ .

$f (- 3) = 0$

Step 6.1.2.6

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = - 1$ .

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Step 6.2.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Combine the numerators over the common denominator.

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

Step 6.2.2.2

Simplify each term.

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Step 6.2.2.2.1

Move $- 1$ to the left of $π$ .

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

Step 6.2.2.2.2

Rewrite $- 1 π$ as $- π$ .

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

Step 6.2.2.3

Simplify terms.

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Step 6.2.2.3.1

Add $- π$ and $3 π$ .

$f (- 1) = 3 \sin (\frac{2 π}{4})$

Step 6.2.2.3.2

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

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Step 6.2.2.3.2.1

Factor $2$ out of $2 π$ .

$f (- 1) = 3 \sin (\frac{2 (π)}{4})$

Step 6.2.2.3.2.2

Cancel the common factors.

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Step 6.2.2.3.2.2.1

Factor $2$ out of $4$ .

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

Step 6.2.2.3.2.2.2

Cancel the common factor.

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

Step 6.2.2.3.2.2.3

Rewrite the expression.

$f (- 1) = 3 \sin (\frac{π}{2})$

Step 6.2.2.4

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

$f (- 1) = 3 \cdot 1$

Step 6.2.2.5

Multiply $3$ by $1$ .

$f (- 1) = 3$

Step 6.2.2.6

The final answer is $3$ .

$3$

Step 6.3

Find the point at $x = 1$ .

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Step 6.3.1

Replace the variable $x$ with $1$ in the expression.

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Combine the numerators over the common denominator.

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

Step 6.3.2.2

Multiply $π$ by $1$ .

$f (1) = 3 \sin (\frac{π + 3 π}{4})$

Step 6.3.2.3

Add $π$ and $3 π$ .

$f (1) = 3 \sin (\frac{4 π}{4})$

Step 6.3.2.4

Cancel the common factor of $4$ .

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Step 6.3.2.4.1

Cancel the common factor.

$f (1) = 3 \sin (\frac{4 π}{4})$

Step 6.3.2.4.2

Divide $π$ by $1$ .

$f (1) = 3 \sin (π)$

Step 6.3.2.5

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

$f (1) = 3 \sin (0)$

Step 6.3.2.6

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

$f (1) = 3 \cdot 0$

Step 6.3.2.7

Multiply $3$ by $0$ .

$f (1) = 0$

Step 6.3.2.8

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = 3$ .

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Step 6.4.1

Replace the variable $x$ with $3$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Combine the numerators over the common denominator.

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

Step 6.4.2.2

Move $3$ to the left of $π$ .

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

Step 6.4.2.3

Simplify terms.

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Step 6.4.2.3.1

Add $3 π$ and $3 π$ .

$f (3) = 3 \sin (\frac{6 π}{4})$

Step 6.4.2.3.2

Cancel the common factor of $6$ and $4$ .

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Step 6.4.2.3.2.1

Factor $2$ out of $6 π$ .

$f (3) = 3 \sin (\frac{2 (3 π)}{4})$

Step 6.4.2.3.2.2

Cancel the common factors.

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Step 6.4.2.3.2.2.1

Factor $2$ out of $4$ .

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

Step 6.4.2.3.2.2.2

Cancel the common factor.

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

Step 6.4.2.3.2.2.3

Rewrite the expression.

$f (3) = 3 \sin (\frac{3 π}{2})$

Step 6.4.2.4

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.

$f (3) = 3 (- \sin (\frac{π}{2}))$

Step 6.4.2.5

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

$f (3) = 3 (- 1 \cdot 1)$

Step 6.4.2.6

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

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Step 6.4.2.6.1

Multiply $- 1$ by $1$ .

$f (3) = 3 \cdot - 1$

Step 6.4.2.6.2

Multiply $3$ by $- 1$ .

$f (3) = - 3$

Step 6.4.2.7

The final answer is $- 3$ .

$- 3$

Step 6.5

Find the point at $x = 5$ .

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Step 6.5.1

Replace the variable $x$ with $5$ in the expression.

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Combine the numerators over the common denominator.

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

Step 6.5.2.2

Move $5$ to the left of $π$ .

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

Step 6.5.2.3

Simplify terms.

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Step 6.5.2.3.1

Add $5 π$ and $3 π$ .

$f (5) = 3 \sin (\frac{8 π}{4})$

Step 6.5.2.3.2

Cancel the common factor of $8$ and $4$ .

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Step 6.5.2.3.2.1

Factor $4$ out of $8 π$ .

$f (5) = 3 \sin (\frac{4 (2 π)}{4})$

Step 6.5.2.3.2.2

Cancel the common factors.

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Step 6.5.2.3.2.2.1

Factor $4$ out of $4$ .

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

Step 6.5.2.3.2.2.2

Cancel the common factor.

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

Step 6.5.2.3.2.2.3

Rewrite the expression.

$f (5) = 3 \sin (\frac{2 π}{1})$

Step 6.5.2.3.2.2.4

Divide $2 π$ by $1$ .

$f (5) = 3 \sin (2 π)$

Step 6.5.2.4

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

$f (5) = 3 \sin (0)$

Step 6.5.2.5

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

$f (5) = 3 \cdot 0$

Step 6.5.2.6

Multiply $3$ by $0$ .

$f (5) = 0$

Step 6.5.2.7

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 3 & 0 \\ - 1 & 3 \\ 1 & 0 \\ 3 & - 3 \\ 5 & 0 \end{array}$

Step 7

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

Amplitude: $3$

Period: $8$

Phase Shift: $- 3$ ( $3$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 3 & 0 \\ - 1 & 3 \\ 1 & 0 \\ 3 & - 3 \\ 5 & 0 \end{array}$

Step 8