Trigonometry Examples

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

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.

$a = 3$

$b = \frac{π}{3}$

$c = - (\frac{2 π}{3} + 1)$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

Find the period of $3 \sin (\frac{π x}{3} + \frac{2 π}{3} + 1)$ .

Tap for more steps...

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 $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

Step 3.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 3.5

Cancel the common factor of $π$ .

Tap for more steps...

Step 3.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 3.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 3.5.3

Rewrite the expression.

$2 \cdot 3$

Step 3.6

Multiply $2$ by $3$ .

$6$

Step 4

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

Tap for more steps...

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: $\frac{- (\frac{2 π}{3} + 1)}{\frac{π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- (\frac{2 π}{3} + 1) \frac{3}{π}$

Step 4.4

Apply the distributive property.

Phase Shift: $(- \frac{2 π}{3} - 1 \cdot 1) (\frac{3}{π})$

Step 4.5

Multiply $- 1$ by $1$ .

Phase Shift: $(- \frac{2 π}{3} - 1) (\frac{3}{π})$

Step 4.6

Apply the distributive property.

Phase Shift: $- \frac{2 π}{3} \cdot \frac{3}{π} - 1 \frac{3}{π}$

Step 4.7

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.7.1

Move the leading negative in $- \frac{2 π}{3}$ into the numerator.

Phase Shift: $\frac{- 2 π}{3} \cdot \frac{3}{π} - 1 \frac{3}{π}$

Step 4.7.2

Factor $π$ out of $- 2 π$ .

Phase Shift: $\frac{π \cdot - 2}{3} \cdot \frac{3}{π} - 1 \frac{3}{π}$

Step 4.7.3

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 2}{3} \cdot \frac{3}{π} - 1 \frac{3}{π}$

Step 4.7.4

Rewrite the expression.

Phase Shift: $\frac{- 2}{3} \cdot 3 - 1 \frac{3}{π}$

Step 4.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.8.1

Cancel the common factor.

Phase Shift: $\frac{- 2}{3} \cdot 3 - 1 \frac{3}{π}$

Step 4.8.2

Rewrite the expression.

Phase Shift: $- 2 - 1 \frac{3}{π}$

Step 4.9

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

Phase Shift: $- 2 - \frac{3}{π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $6$

Phase Shift: $- 2 - \frac{3}{π}$ ( $- (- 2 - \frac{3}{π})$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

Simplify the result.

Tap for more steps...

Step 6.1.2.1

Combine the numerators over the common denominator.

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

Step 6.1.2.2

Simplify each term.

Tap for more steps...

Step 6.1.2.2.1

Apply the distributive property.

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

Step 6.1.2.2.2

Move $- 2$ to the left of $π$ .

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

Step 6.1.2.2.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.1.2.2.3.1

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

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

Step 6.1.2.2.3.2

Cancel the common factor.

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

Step 6.1.2.2.3.3

Rewrite the expression.

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

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

Step 6.1.2.3

Simplify by adding terms.

Tap for more steps...

Step 6.1.2.3.1

Add $- 2 π$ and $2 π$ .

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

Step 6.1.2.3.2

Simplify the expression.

Tap for more steps...

Step 6.1.2.3.2.1

Subtract $3$ from $0$ .

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

Step 6.1.2.3.2.2

Divide $- 3$ by $3$ .

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

Step 6.1.2.3.2.3

Subtract $1$ from $1$ .

$f (- 2 - \frac{3}{π}) = 3 \sin (0)$

Step 6.1.2.4

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

$f (- 2 - \frac{3}{π}) = 3 \cdot 0$

Step 6.1.2.5

Multiply $3$ by $0$ .

$f (- 2 - \frac{3}{π}) = 0$

Step 6.1.2.6

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = - \frac{1}{2} - \frac{3}{π}$ .

Tap for more steps...

Step 6.2.1

Replace the variable $x$ with $- \frac{1}{2} - \frac{3}{π}$ in the expression.

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

Step 6.2.2

Simplify the result.

Tap for more steps...

Step 6.2.2.1

Combine the numerators over the common denominator.

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

Step 6.2.2.2

Simplify each term.

Tap for more steps...

Step 6.2.2.2.1

Apply the distributive property.

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

Step 6.2.2.2.2

Combine $π$ and $\frac{1}{2}$ .

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

Step 6.2.2.2.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.2.2.2.3.1

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

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

Step 6.2.2.2.3.2

Cancel the common factor.

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

Step 6.2.2.2.3.3

Rewrite the expression.

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

Step 6.2.2.3

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

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

Step 6.2.2.4

Combine fractions.

Tap for more steps...

Step 6.2.2.4.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.2.2.4.2

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Simplify the numerator.

Tap for more steps...

Step 6.2.2.5.1

Multiply $2$ by $2$ .

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

Step 6.2.2.5.2

Add $- π$ and $4 π$ .

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

Step 6.2.2.6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.2.2.6.1

Cancel the common factor of $\frac{3 π}{2} - 3$ and $3$ .

Tap for more steps...

Step 6.2.2.6.1.1

Factor $3$ out of $\frac{3 π}{2}$ .

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

Step 6.2.2.6.1.2

Factor $3$ out of $- 3$ .

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

Step 6.2.2.6.1.3

Factor $3$ out of $3 \frac{π}{2} + 3 (- 1)$ .

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

Step 6.2.2.6.1.4

Cancel the common factors.

Tap for more steps...

Step 6.2.2.6.1.4.1

Factor $3$ out of $3$ .

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

Step 6.2.2.6.1.4.2

Cancel the common factor.

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

Step 6.2.2.6.1.4.3

Rewrite the expression.

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

Step 6.2.2.6.1.4.4

Divide $\frac{π}{2} - 1$ by $1$ .

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

Step 6.2.2.6.2

Simplify by subtracting numbers.

Tap for more steps...

Step 6.2.2.6.2.1

Subtract $1$ from $1$ .

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

Step 6.2.2.6.2.2

Add $0$ and $\frac{π}{2}$ .

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

Step 6.2.2.7

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

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

Step 6.2.2.8

Multiply $3$ by $1$ .

$f (- \frac{1}{2} - \frac{3}{π}) = 3$

Step 6.2.2.9

The final answer is $3$ .

$3$

Step 6.3

Find the point at $x = 1 - \frac{3}{π}$ .

Tap for more steps...

Step 6.3.1

Replace the variable $x$ with $1 - \frac{3}{π}$ in the expression.

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

Step 6.3.2

Simplify the result.

Tap for more steps...

Step 6.3.2.1

Combine the numerators over the common denominator.

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

Step 6.3.2.2

Simplify each term.

Tap for more steps...

Step 6.3.2.2.1

Apply the distributive property.

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

Step 6.3.2.2.2

Multiply $π$ by $1$ .

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

Step 6.3.2.2.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.3.2.2.3.1

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

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

Step 6.3.2.2.3.2

Cancel the common factor.

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

Step 6.3.2.2.3.3

Rewrite the expression.

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

Step 6.3.2.3

Simplify terms.

Tap for more steps...

Step 6.3.2.3.1

Add $π$ and $2 π$ .

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

Step 6.3.2.3.2

Cancel the common factor of $3 π - 3$ and $3$ .

Tap for more steps...

Step 6.3.2.3.2.1

Factor $3$ out of $3 π$ .

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

Step 6.3.2.3.2.2

Factor $3$ out of $- 3$ .

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

Step 6.3.2.3.2.3

Factor $3$ out of $3 (π) + 3 (- 1)$ .

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

Step 6.3.2.3.2.4

Cancel the common factors.

Tap for more steps...

Step 6.3.2.3.2.4.1

Factor $3$ out of $3$ .

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

Step 6.3.2.3.2.4.2

Cancel the common factor.

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

Step 6.3.2.3.2.4.3

Rewrite the expression.

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

Step 6.3.2.3.2.4.4

Divide $π - 1$ by $1$ .

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

Step 6.3.2.3.3

Simplify by subtracting numbers.

Tap for more steps...

Step 6.3.2.3.3.1

Subtract $1$ from $1$ .

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

Step 6.3.2.3.3.2

Add $0$ and $π$ .

$f (1 - \frac{3}{π}) = 3 \sin (π)$

Step 6.3.2.4

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

$f (1 - \frac{3}{π}) = 3 \sin (0)$

Step 6.3.2.5

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

$f (1 - \frac{3}{π}) = 3 \cdot 0$

Step 6.3.2.6

Multiply $3$ by $0$ .

$f (1 - \frac{3}{π}) = 0$

Step 6.3.2.7

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{5}{2} - \frac{3}{π}$ .

Tap for more steps...

Step 6.4.1

Replace the variable $x$ with $\frac{5}{2} - \frac{3}{π}$ in the expression.

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

Step 6.4.2

Simplify the result.

Tap for more steps...

Step 6.4.2.1

Combine the numerators over the common denominator.

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

Step 6.4.2.2

Simplify each term.

Tap for more steps...

Step 6.4.2.2.1

Apply the distributive property.

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

Step 6.4.2.2.2

Combine $π$ and $\frac{5}{2}$ .

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

Step 6.4.2.2.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.4.2.2.3.1

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

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

Step 6.4.2.2.3.2

Cancel the common factor.

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

Step 6.4.2.2.3.3

Rewrite the expression.

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

Step 6.4.2.2.4

Move $5$ to the left of $π$ .

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

Step 6.4.2.3

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

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

Step 6.4.2.4

Combine fractions.

Tap for more steps...

Step 6.4.2.4.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.4.2.4.2

Combine the numerators over the common denominator.

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

Step 6.4.2.5

Simplify the numerator.

Tap for more steps...

Step 6.4.2.5.1

Multiply $2$ by $2$ .

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

Step 6.4.2.5.2

Add $5 π$ and $4 π$ .

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

Step 6.4.2.6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.4.2.6.1

Cancel the common factor of $\frac{9 π}{2} - 3$ and $3$ .

Tap for more steps...

Step 6.4.2.6.1.1

Factor $3$ out of $\frac{9 π}{2}$ .

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

Step 6.4.2.6.1.2

Factor $3$ out of $- 3$ .

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

Step 6.4.2.6.1.3

Factor $3$ out of $3 \frac{3 π}{2} + 3 (- 1)$ .

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

Step 6.4.2.6.1.4

Cancel the common factors.

Tap for more steps...

Step 6.4.2.6.1.4.1

Factor $3$ out of $3$ .

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

Step 6.4.2.6.1.4.2

Cancel the common factor.

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

Step 6.4.2.6.1.4.3

Rewrite the expression.

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

Step 6.4.2.6.1.4.4

Divide $\frac{3 π}{2} - 1$ by $1$ .

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

Step 6.4.2.6.2

Simplify by subtracting numbers.

Tap for more steps...

Step 6.4.2.6.2.1

Subtract $1$ from $1$ .

$f (\frac{5}{2} - \frac{3}{π}) = 3 \sin (0 + \frac{3 π}{2})$

Step 6.4.2.6.2.2

Add $0$ and $\frac{3 π}{2}$ .

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

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.

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

Step 6.4.2.8

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

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

Step 6.4.2.9

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

Tap for more steps...

Step 6.4.2.9.1

Multiply $- 1$ by $1$ .

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

Step 6.4.2.9.2

Multiply $3$ by $- 1$ .

$f (\frac{5}{2} - \frac{3}{π}) = - 3$

Step 6.4.2.10

The final answer is $- 3$ .

$- 3$

Step 6.5

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

Tap for more steps...

Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

Tap for more steps...

Step 6.5.2.1

Combine the numerators over the common denominator.

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

Step 6.5.2.2

Simplify each term.

Tap for more steps...

Step 6.5.2.2.1

Apply the distributive property.

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

Step 6.5.2.2.2

Move $4$ to the left of $π$ .

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

Step 6.5.2.2.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 6.5.2.2.3.1

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

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

Step 6.5.2.2.3.2

Cancel the common factor.

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

Step 6.5.2.2.3.3

Rewrite the expression.

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

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

Step 6.5.2.3

Simplify terms.

Tap for more steps...

Step 6.5.2.3.1

Add $4 π$ and $2 π$ .

$f (4 - \frac{3}{π}) = 3 \sin (1 + \frac{6 π - 3}{3})$

Step 6.5.2.3.2

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

Tap for more steps...

Step 6.5.2.3.2.1

Factor $3$ out of $6 π$ .

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

Step 6.5.2.3.2.2

Factor $3$ out of $- 3$ .

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

Step 6.5.2.3.2.3

Factor $3$ out of $3 (2 π) + 3 (- 1)$ .

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

Step 6.5.2.3.2.4

Cancel the common factors.

Tap for more steps...

Step 6.5.2.3.2.4.1

Factor $3$ out of $3$ .

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

Step 6.5.2.3.2.4.2

Cancel the common factor.

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

Step 6.5.2.3.2.4.3

Rewrite the expression.

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

Step 6.5.2.3.2.4.4

Divide $2 π - 1$ by $1$ .

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

Step 6.5.2.3.3

Simplify by subtracting numbers.

Tap for more steps...

Step 6.5.2.3.3.1

Subtract $1$ from $1$ .

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

Step 6.5.2.3.3.2

Add $2 π$ and $0$ .

$f (4 - \frac{3}{π}) = 3 \sin (2 π)$

Step 6.5.2.4

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

$f (4 - \frac{3}{π}) = 3 \sin (0)$

Step 6.5.2.5

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

$f (4 - \frac{3}{π}) = 3 \cdot 0$

Step 6.5.2.6

Multiply $3$ by $0$ .

$f (4 - \frac{3}{π}) = 0$

Step 6.5.2.7

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 2 - \frac{3}{π} & 0 \\ - \frac{1}{2} - \frac{3}{π} & 3 \\ 1 - \frac{3}{π} & 0 \\ \frac{5}{2} - \frac{3}{π} & - 3 \\ 4 - \frac{3}{π} & 0 \end{array}$

Step 7

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

Amplitude: $3$

Period: $6$

Phase Shift: $- 2 - \frac{3}{π}$ ( $- (- 2 - \frac{3}{π})$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - 2 - \frac{3}{π} & 0 \\ - \frac{1}{2} - \frac{3}{π} & 3 \\ 1 - \frac{3}{π} & 0 \\ \frac{5}{2} - \frac{3}{π} & - 3 \\ 4 - \frac{3}{π} & 0 \end{array}$

Step 8