Trigonometry Examples

$f (x) = - 3 \sin (\frac{π x}{10} - \frac{2 π}{5}) - 1$

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.

$a = - 3$

$b = \frac{π}{10}$

$c = \frac{2 π}{5}$

$d = - 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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Step 3.1

Find the period of $- 3 \sin (\frac{π x}{10} - \frac{2 π}{5})$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{π}{10}$ in the formula for period.

$\frac{2 π}{| \frac{π}{10} |}$

Step 3.1.3

$\frac{π}{10}$ is approximately $0.31415926$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{10}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{10}{π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{10}{π}$

Step 3.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{10}{π}$

Step 3.1.5.3

Rewrite the expression.

$2 \cdot 10$

Step 3.1.6

Multiply $2$ by $10$ .

$20$

Step 3.2

Find the period of $- 1$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{π}{10}$ in the formula for period.

$\frac{2 π}{| \frac{π}{10} |}$

Step 3.2.3

$\frac{π}{10}$ is approximately $0.31415926$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{10}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{10}{π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{10}{π}$

Step 3.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{10}{π}$

Step 3.2.5.3

Rewrite the expression.

$2 \cdot 10$

Step 3.2.6

Multiply $2$ by $10$ .

$20$

Step 3.3

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

$20$

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: $\frac{\frac{2 π}{5}}{\frac{π}{10}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{2 π}{5} \cdot \frac{10}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Factor $π$ out of $2 π$ .

Phase Shift: $\frac{π \cdot 2}{5} \cdot \frac{10}{π}$

Step 4.4.2

Cancel the common factor.

Phase Shift: $\frac{π \cdot 2}{5} \cdot \frac{10}{π}$

Step 4.4.3

Rewrite the expression.

Phase Shift: $\frac{2}{5} \cdot 10$

Step 4.5

Cancel the common factor of $5$ .

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Step 4.5.1

Factor $5$ out of $10$ .

Phase Shift: $\frac{2}{5} \cdot (5 (2))$

Step 4.5.2

Cancel the common factor.

Phase Shift: $\frac{2}{5} \cdot (5 \cdot 2)$

Step 4.5.3

Rewrite the expression.

Phase Shift: $2 \cdot 2$

Step 4.6

Multiply $2$ by $2$ .

Phase Shift: $4$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $20$

Phase Shift: $4$ ( $4$ to the right)

Vertical Shift: $- 1$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 4$ .

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Step 6.1.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = - 3 \sin (\frac{π (4)}{10} - \frac{2 π}{5}) - 1$

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

Simplify each term.

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Step 6.1.2.1.1.1

Cancel the common factor of $4$ and $10$ .

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Step 6.1.2.1.1.1.1

Factor $2$ out of $π (4)$ .

$f (4) = - 3 \sin (\frac{2 (π \cdot (2))}{10} - \frac{2 π}{5}) - 1$

Step 6.1.2.1.1.1.2

Cancel the common factors.

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Step 6.1.2.1.1.1.2.1

Factor $2$ out of $10$ .

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

Step 6.1.2.1.1.1.2.2

Cancel the common factor.

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

Step 6.1.2.1.1.1.2.3

Rewrite the expression.

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

Step 6.1.2.1.1.2

Move $2$ to the left of $π$ .

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

Step 6.1.2.1.2

Combine the numerators over the common denominator.

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

Step 6.1.2.1.3

Subtract $2 π$ from $2 π$ .

$f (4) = - 3 \sin (\frac{0}{5}) - 1$

Step 6.1.2.1.4

Divide $0$ by $5$ .

$f (4) = - 3 \sin (0) - 1$

Step 6.1.2.1.5

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

$f (4) = - 3 \cdot 0 - 1$

Step 6.1.2.1.6

Multiply $- 3$ by $0$ .

$f (4) = 0 - 1$

Step 6.1.2.2

Subtract $1$ from $0$ .

$f (4) = - 1$

Step 6.1.2.3

The final answer is $- 1$ .

$- 1$

Step 6.2

Find the point at $x = 9$ .

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Step 6.2.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = - 3 \sin (\frac{π (9)}{10} - \frac{2 π}{5}) - 1$

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

Move $9$ to the left of $π$ .

$f (9) = - 3 \sin (\frac{9 π}{10} - \frac{2 π}{5}) - 1$

Step 6.2.2.1.2

To write $- \frac{2 π}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (9) = - 3 \sin (\frac{9 π}{10} - \frac{2 π}{5} \cdot \frac{2}{2}) - 1$

Step 6.2.2.1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.1.3.1

Multiply $\frac{2 π}{5}$ by $\frac{2}{2}$ .

$f (9) = - 3 \sin (\frac{9 π}{10} - \frac{2 π \cdot 2}{5 \cdot 2}) - 1$

Step 6.2.2.1.3.2

Multiply $5$ by $2$ .

$f (9) = - 3 \sin (\frac{9 π}{10} - \frac{2 π \cdot 2}{10}) - 1$

Step 6.2.2.1.4

Combine the numerators over the common denominator.

$f (9) = - 3 \sin (\frac{9 π - 2 π \cdot 2}{10}) - 1$

Step 6.2.2.1.5

Simplify the numerator.

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Step 6.2.2.1.5.1

Multiply $2$ by $- 2$ .

$f (9) = - 3 \sin (\frac{9 π - 4 π}{10}) - 1$

Step 6.2.2.1.5.2

Subtract $4 π$ from $9 π$ .

$f (9) = - 3 \sin (\frac{5 π}{10}) - 1$

Step 6.2.2.1.6

Cancel the common factor of $5$ and $10$ .

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Step 6.2.2.1.6.1

Factor $5$ out of $5 π$ .

$f (9) = - 3 \sin (\frac{5 (π)}{10}) - 1$

Step 6.2.2.1.6.2

Cancel the common factors.

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Step 6.2.2.1.6.2.1

Factor $5$ out of $10$ .

$f (9) = - 3 \sin (\frac{5 π}{5 \cdot 2}) - 1$

Step 6.2.2.1.6.2.2

Cancel the common factor.

$f (9) = - 3 \sin (\frac{5 π}{5 \cdot 2}) - 1$

Step 6.2.2.1.6.2.3

Rewrite the expression.

$f (9) = - 3 \sin (\frac{π}{2}) - 1$

Step 6.2.2.1.7

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

$f (9) = - 3 \cdot 1 - 1$

Step 6.2.2.1.8

Multiply $- 3$ by $1$ .

$f (9) = - 3 - 1$

Step 6.2.2.2

Subtract $1$ from $- 3$ .

$f (9) = - 4$

Step 6.2.2.3

The final answer is $- 4$ .

$- 4$

Step 6.3

Find the point at $x = 14$ .

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Step 6.3.1

Replace the variable $x$ with $14$ in the expression.

$f (14) = - 3 \sin (\frac{π (14)}{10} - \frac{2 π}{5}) - 1$

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

Simplify each term.

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Step 6.3.2.1.1.1

Cancel the common factor of $14$ and $10$ .

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Step 6.3.2.1.1.1.1

Factor $2$ out of $π (14)$ .

$f (14) = - 3 \sin (\frac{2 (π \cdot (7))}{10} - \frac{2 π}{5}) - 1$

Step 6.3.2.1.1.1.2

Cancel the common factors.

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Step 6.3.2.1.1.1.2.1

Factor $2$ out of $10$ .

$f (14) = - 3 \sin (\frac{2 (π \cdot (7))}{2 (5)} - \frac{2 π}{5}) - 1$

Step 6.3.2.1.1.1.2.2

Cancel the common factor.

$f (14) = - 3 \sin (\frac{2 (π \cdot (7))}{2 \cdot 5} - \frac{2 π}{5}) - 1$

Step 6.3.2.1.1.1.2.3

Rewrite the expression.

$f (14) = - 3 \sin (\frac{π \cdot (7)}{5} - \frac{2 π}{5}) - 1$

Step 6.3.2.1.1.2

Move $7$ to the left of $π$ .

$f (14) = - 3 \sin (\frac{7 π}{5} - \frac{2 π}{5}) - 1$

Step 6.3.2.1.2

Combine the numerators over the common denominator.

$f (14) = - 3 \sin (\frac{7 π - 2 π}{5}) - 1$

Step 6.3.2.1.3

Subtract $2 π$ from $7 π$ .

$f (14) = - 3 \sin (\frac{5 π}{5}) - 1$

Step 6.3.2.1.4

Cancel the common factor of $5$ .

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Step 6.3.2.1.4.1

Cancel the common factor.

$f (14) = - 3 \sin (\frac{5 π}{5}) - 1$

Step 6.3.2.1.4.2

Divide $π$ by $1$ .

$f (14) = - 3 \sin (π) - 1$

Step 6.3.2.1.5

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

$f (14) = - 3 \sin (0) - 1$

Step 6.3.2.1.6

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

$f (14) = - 3 \cdot 0 - 1$

Step 6.3.2.1.7

Multiply $- 3$ by $0$ .

$f (14) = 0 - 1$

Step 6.3.2.2

Subtract $1$ from $0$ .

$f (14) = - 1$

Step 6.3.2.3

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = 19$ .

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Step 6.4.1

Replace the variable $x$ with $19$ in the expression.

$f (19) = - 3 \sin (\frac{π (19)}{10} - \frac{2 π}{5}) - 1$

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

Move $19$ to the left of $π$ .

$f (19) = - 3 \sin (\frac{19 π}{10} - \frac{2 π}{5}) - 1$

Step 6.4.2.1.2

To write $- \frac{2 π}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (19) = - 3 \sin (\frac{19 π}{10} - \frac{2 π}{5} \cdot \frac{2}{2}) - 1$

Step 6.4.2.1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.1.3.1

Multiply $\frac{2 π}{5}$ by $\frac{2}{2}$ .

$f (19) = - 3 \sin (\frac{19 π}{10} - \frac{2 π \cdot 2}{5 \cdot 2}) - 1$

Step 6.4.2.1.3.2

Multiply $5$ by $2$ .

$f (19) = - 3 \sin (\frac{19 π}{10} - \frac{2 π \cdot 2}{10}) - 1$

Step 6.4.2.1.4

Combine the numerators over the common denominator.

$f (19) = - 3 \sin (\frac{19 π - 2 π \cdot 2}{10}) - 1$

Step 6.4.2.1.5

Simplify the numerator.

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Step 6.4.2.1.5.1

Multiply $2$ by $- 2$ .

$f (19) = - 3 \sin (\frac{19 π - 4 π}{10}) - 1$

Step 6.4.2.1.5.2

Subtract $4 π$ from $19 π$ .

$f (19) = - 3 \sin (\frac{15 π}{10}) - 1$

Step 6.4.2.1.6

Cancel the common factor of $15$ and $10$ .

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Step 6.4.2.1.6.1

Factor $5$ out of $15 π$ .

$f (19) = - 3 \sin (\frac{5 (3 π)}{10}) - 1$

Step 6.4.2.1.6.2

Cancel the common factors.

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Step 6.4.2.1.6.2.1

Factor $5$ out of $10$ .

$f (19) = - 3 \sin (\frac{5 (3 π)}{5 (2)}) - 1$

Step 6.4.2.1.6.2.2

Cancel the common factor.

$f (19) = - 3 \sin (\frac{5 (3 π)}{5 \cdot 2}) - 1$

Step 6.4.2.1.6.2.3

Rewrite the expression.

$f (19) = - 3 \sin (\frac{3 π}{2}) - 1$

Step 6.4.2.1.7

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.

$f (19) = - 3 (- \sin (\frac{π}{2})) - 1$

Step 6.4.2.1.8

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

$f (19) = - 3 (- 1 \cdot 1) - 1$

Step 6.4.2.1.9

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

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Step 6.4.2.1.9.1

Multiply $- 1$ by $1$ .

$f (19) = - 3 \cdot - 1 - 1$

Step 6.4.2.1.9.2

Multiply $- 3$ by $- 1$ .

$f (19) = 3 - 1$

Step 6.4.2.2

Subtract $1$ from $3$ .

$f (19) = 2$

Step 6.4.2.3

The final answer is $2$ .

$2$

Step 6.5

Find the point at $x = 24$ .

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Step 6.5.1

Replace the variable $x$ with $24$ in the expression.

$f (24) = - 3 \sin (\frac{π (24)}{10} - \frac{2 π}{5}) - 1$

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

Simplify each term.

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Step 6.5.2.1.1.1

Cancel the common factor of $24$ and $10$ .

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Step 6.5.2.1.1.1.1

Factor $2$ out of $π (24)$ .

$f (24) = - 3 \sin (\frac{2 (π \cdot (12))}{10} - \frac{2 π}{5}) - 1$

Step 6.5.2.1.1.1.2

Cancel the common factors.

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Step 6.5.2.1.1.1.2.1

Factor $2$ out of $10$ .

$f (24) = - 3 \sin (\frac{2 (π \cdot (12))}{2 (5)} - \frac{2 π}{5}) - 1$

Step 6.5.2.1.1.1.2.2

Cancel the common factor.

$f (24) = - 3 \sin (\frac{2 (π \cdot (12))}{2 \cdot 5} - \frac{2 π}{5}) - 1$

Step 6.5.2.1.1.1.2.3

Rewrite the expression.

$f (24) = - 3 \sin (\frac{π \cdot (12)}{5} - \frac{2 π}{5}) - 1$

Step 6.5.2.1.1.2

Move $12$ to the left of $π$ .

$f (24) = - 3 \sin (\frac{12 π}{5} - \frac{2 π}{5}) - 1$

Step 6.5.2.1.2

Combine the numerators over the common denominator.

$f (24) = - 3 \sin (\frac{12 π - 2 π}{5}) - 1$

Step 6.5.2.1.3

Subtract $2 π$ from $12 π$ .

$f (24) = - 3 \sin (\frac{10 π}{5}) - 1$

Step 6.5.2.1.4

Cancel the common factor of $10$ and $5$ .

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Step 6.5.2.1.4.1

Factor $5$ out of $10 π$ .

$f (24) = - 3 \sin (\frac{5 (2 π)}{5}) - 1$

Step 6.5.2.1.4.2

Cancel the common factors.

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Step 6.5.2.1.4.2.1

Factor $5$ out of $5$ .

$f (24) = - 3 \sin (\frac{5 (2 π)}{5 (1)}) - 1$

Step 6.5.2.1.4.2.2

Cancel the common factor.

$f (24) = - 3 \sin (\frac{5 (2 π)}{5 \cdot 1}) - 1$

Step 6.5.2.1.4.2.3

Rewrite the expression.

$f (24) = - 3 \sin (\frac{2 π}{1}) - 1$

Step 6.5.2.1.4.2.4

Divide $2 π$ by $1$ .

$f (24) = - 3 \sin (2 π) - 1$

Step 6.5.2.1.5

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

$f (24) = - 3 \sin (0) - 1$

Step 6.5.2.1.6

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

$f (24) = - 3 \cdot 0 - 1$

Step 6.5.2.1.7

Multiply $- 3$ by $0$ .

$f (24) = 0 - 1$

Step 6.5.2.2

Subtract $1$ from $0$ .

$f (24) = - 1$

Step 6.5.2.3

The final answer is $- 1$ .

$- 1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 4 & - 1 \\ 9 & - 4 \\ 14 & - 1 \\ 19 & 2 \\ 24 & - 1 \end{array}$

Step 7

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

Amplitude: $3$

Period: $20$

Phase Shift: $4$ ( $4$ to the right)

Vertical Shift: $- 1$

$\begin{array}{c} x & f (x) \\ 4 & - 1 \\ 9 & - 4 \\ 14 & - 1 \\ 19 & 2 \\ 24 & - 1 \end{array}$

Step 8