Trigonometry Examples

$y = \frac{- 1}{2} \cdot \cos (\frac{2 π}{5} x - π)$

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

$a = - \frac{1}{2}$

$b = \frac{2 π}{5}$

$c = π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{5} |}$

$\frac{2 π}{5}$ is approximately $1.25663706$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Cancel the common factor of $2 π$ .

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Cancel the common factor.

$2 π \frac{5}{2 π}$

Rewrite the expression.

$5$

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

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

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

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

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

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{5}{2 π})$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $5$

Phase Shift: $\frac{5}{2}$ ( $\frac{5}{2}$ to the right)

Vertical Shift: None

Select a few points to graph.

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

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

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

Simplify the result.

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Simplify the numerator.

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Combine $\frac{5}{2}$ and $2$ .

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

Combine $\frac{5 \cdot 2}{2}$ and $π$ .

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

Multiply $5$ by $2$ .

$f (\frac{5}{2}) = - \frac{\cos (\frac{\frac{10 π}{2}}{5} - π)}{2}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{10 π}{2}$ by cancelling the common factors.

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Factor $2$ out of $10 π$ .

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

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $5 π$ by $1$ .

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

Simplify the numerator.

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Cancel the common factor of $5$ .

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Cancel the common factor.

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

Divide $π$ by $1$ .

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

Subtract $π$ from $π$ .

$f (\frac{5}{2}) = - \frac{\cos (0)}{2}$

The exact value of $\cos (0)$ is $1$ .

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

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Find the point at $x = \frac{15}{4}$ .

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Replace the variable $x$ with $\frac{15}{4}$ in the expression.

$f (\frac{15}{4}) = - \frac{\cos (\frac{2 π (\frac{15}{4})}{5} - π)}{2}$

Simplify the result.

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Simplify the numerator.

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Combine $\frac{15}{4}$ and $2$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{15 \cdot 2}{4} \cdot π}{5} - π)}{2}$

Combine $\frac{15 \cdot 2}{4}$ and $π$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{15 \cdot (2 π)}{4}}{5} - π)}{2}$

Reduce the expression by cancelling the common factors.

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Multiply $15$ by $2$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{30 π}{4}}{5} - π)}{2}$

Reduce the expression $\frac{30 π}{4}$ by cancelling the common factors.

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Factor $2$ out of $30 π$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{2 (15 π)}{4}}{5} - π)}{2}$

Factor $2$ out of $4$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{2 (15 π)}{2 (2)}}{5} - π)}{2}$

Cancel the common factor.

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{2 (15 π)}{2 \cdot 2}}{5} - π)}{2}$

Rewrite the expression.

$f (\frac{15}{4}) = - \frac{\cos (\frac{\frac{15 π}{2}}{5} - π)}{2}$

Simplify the numerator.

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

$f (\frac{15}{4}) = - \frac{\cos (\frac{15 π}{2} \cdot \frac{1}{5} - π)}{2}$

Cancel the common factor of $5$ .

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Factor $5$ out of $15 π$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{5 (3 π)}{2} \cdot \frac{1}{5} - π)}{2}$

Cancel the common factor.

$f (\frac{15}{4}) = - \frac{\cos (\frac{5 (3 π)}{2} \cdot \frac{1}{5} - π)}{2}$

Rewrite the expression.

$f (\frac{15}{4}) = - \frac{\cos (\frac{3 π}{2} - π)}{2}$

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

$f (\frac{15}{4}) = - \frac{\cos (\frac{3 π}{2} - π \cdot \frac{2}{2})}{2}$

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{3 π}{2} + \frac{- π \cdot 2}{2})}{2}$

Combine the numerators over the common denominator.

$f (\frac{15}{4}) = - \frac{\cos (\frac{3 π - π \cdot 2}{2})}{2}$

Simplify the numerator.

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Multiply $2$ by $- 1$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{3 π - 2 π}{2})}{2}$

Subtract $2 π$ from $3 π$ .

$f (\frac{15}{4}) = - \frac{\cos (\frac{π}{2})}{2}$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{15}{4}) = - \frac{0}{2}$

Simplify the expression.

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Divide $0$ by $2$ .

$f (\frac{15}{4}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{15}{4}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 5$ .

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

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

Simplify the result.

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Cancel the common factor of $5$ .

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Cancel the common factor.

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

Divide $2 π$ by $1$ .

$f (5) = - \frac{\cos (2 π - π)}{2}$

Simplify the numerator.

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Subtract $π$ from $2 π$ .

$f (5) = - \frac{\cos (π)}{2}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (5) = - \frac{- \cos (0)}{2}$

The exact value of $\cos (0)$ is $1$ .

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

Multiply $- 1$ by $1$ .

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

Move the negative in front of the fraction.

$f (5) = \frac{1}{2}$

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Find the point at $x = \frac{25}{4}$ .

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Replace the variable $x$ with $\frac{25}{4}$ in the expression.

$f (\frac{25}{4}) = - \frac{\cos (\frac{2 π (\frac{25}{4})}{5} - π)}{2}$

Simplify the result.

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Simplify the numerator.

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Combine $\frac{25}{4}$ and $2$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{25 \cdot 2}{4} \cdot π}{5} - π)}{2}$

Combine $\frac{25 \cdot 2}{4}$ and $π$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{25 \cdot (2 π)}{4}}{5} - π)}{2}$

Reduce the expression by cancelling the common factors.

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Multiply $25$ by $2$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{50 π}{4}}{5} - π)}{2}$

Reduce the expression $\frac{50 π}{4}$ by cancelling the common factors.

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Factor $2$ out of $50 π$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{2 (25 π)}{4}}{5} - π)}{2}$

Factor $2$ out of $4$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{2 (25 π)}{2 (2)}}{5} - π)}{2}$

Cancel the common factor.

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{2 (25 π)}{2 \cdot 2}}{5} - π)}{2}$

Rewrite the expression.

$f (\frac{25}{4}) = - \frac{\cos (\frac{\frac{25 π}{2}}{5} - π)}{2}$

Simplify the numerator.

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

$f (\frac{25}{4}) = - \frac{\cos (\frac{25 π}{2} \cdot \frac{1}{5} - π)}{2}$

Cancel the common factor of $5$ .

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Factor $5$ out of $25 π$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 (5 π)}{2} \cdot \frac{1}{5} - π)}{2}$

Cancel the common factor.

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 (5 π)}{2} \cdot \frac{1}{5} - π)}{2}$

Rewrite the expression.

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 π}{2} - π)}{2}$

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

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 π}{2} - π \cdot \frac{2}{2})}{2}$

Combine $- π$ and $\frac{2}{2}$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 π}{2} + \frac{- π \cdot 2}{2})}{2}$

Combine the numerators over the common denominator.

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 π - π \cdot 2}{2})}{2}$

Simplify the numerator.

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Multiply $2$ by $- 1$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{5 π - 2 π}{2})}{2}$

Subtract $2 π$ from $5 π$ .

$f (\frac{25}{4}) = - \frac{\cos (\frac{3 π}{2})}{2}$

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

$f (\frac{25}{4}) = - \frac{\cos (\frac{π}{2})}{2}$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{25}{4}) = - \frac{0}{2}$

Simplify the expression.

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Divide $0$ by $2$ .

$f (\frac{25}{4}) = - 0$

Multiply $- 1$ by $0$ .

$f (\frac{25}{4}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{15}{2}$ .

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Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f (\frac{15}{2}) = - \frac{\cos (\frac{2 π (\frac{15}{2})}{5} - π)}{2}$

Simplify the result.

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Simplify the numerator.

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Combine $\frac{15}{2}$ and $2$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{15 \cdot 2}{2} \cdot π}{5} - π)}{2}$

Combine $\frac{15 \cdot 2}{2}$ and $π$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{15 \cdot (2 π)}{2}}{5} - π)}{2}$

Multiply $15$ by $2$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{30 π}{2}}{5} - π)}{2}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{30 π}{2}$ by cancelling the common factors.

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Factor $2$ out of $30 π$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{2 (15 π)}{2}}{5} - π)}{2}$

Factor $2$ out of $2$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{2 (15 π)}{2 (1)}}{5} - π)}{2}$

Cancel the common factor.

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{2 (15 π)}{2 \cdot 1}}{5} - π)}{2}$

Rewrite the expression.

$f (\frac{15}{2}) = - \frac{\cos (\frac{\frac{15 π}{1}}{5} - π)}{2}$

Divide $15 π$ by $1$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{15 π}{5} - π)}{2}$

Simplify the numerator.

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Cancel the common factor of $15$ and $5$ .

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Factor $5$ out of $15 π$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{5 (3 π)}{5} - π)}{2}$

Cancel the common factors.

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Factor $5$ out of $5$ .

$f (\frac{15}{2}) = - \frac{\cos (\frac{5 (3 π)}{5 (1)} - π)}{2}$

Cancel the common factor.

$f (\frac{15}{2}) = - \frac{\cos (\frac{5 (3 π)}{5 \cdot 1} - π)}{2}$

Rewrite the expression.

$f (\frac{15}{2}) = - \frac{\cos (\frac{3 π}{1} - π)}{2}$

Divide $3 π$ by $1$ .

$f (\frac{15}{2}) = - \frac{\cos (3 π - π)}{2}$

Subtract $π$ from $3 π$ .

$f (\frac{15}{2}) = - \frac{\cos (2 π)}{2}$

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

$f (\frac{15}{2}) = - \frac{\cos (0)}{2}$

The exact value of $\cos (0)$ is $1$ .

$f (\frac{15}{2}) = - \frac{1}{2}$

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{5}{2} & - \frac{1}{2} \\ \frac{15}{4} & 0 \\ 5 & \frac{1}{2} \\ \frac{25}{4} & 0 \\ \frac{15}{2} & - \frac{1}{2} \end{array}$

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

Amplitude: $\frac{1}{2}$

Period: $5$

Phase Shift: $\frac{5}{2}$ ( $\frac{5}{2}$ to the right)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ \frac{5}{2} & - \frac{1}{2} \\ \frac{15}{4} & 0 \\ 5 & \frac{1}{2} \\ \frac{25}{4} & 0 \\ \frac{15}{2} & - \frac{1}{2} \end{array}$