Trigonometry Examples

$f (x) = 4 \cos (2 x - π)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 \cos (2 x - π)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 \cos (2 x - π) = 0$ .

$4 \cos (2 x - π) = 0$

Step 1.2.2

Divide each term in $4 \cos (2 x - π) = 0$ by $4$ and simplify.

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Step 1.2.2.1

Divide each term in $4 \cos (2 x - π) = 0$ by $4$ .

$\frac{4 \cos (2 x - π)}{4} = \frac{0}{4}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 \cos (2 x - π)}{4} = \frac{0}{4}$

Step 1.2.2.2.1.2

Divide $\cos (2 x - π)$ by $1$ .

$\cos (2 x - π) = \frac{0}{4}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $4$ .

$\cos (2 x - π) = 0$

Step 1.2.3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$2 x - π = \arccos (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$2 x - π = \frac{π}{2}$

Step 1.2.5

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.1

Add $π$ to both sides of the equation.

$2 x = \frac{π}{2} + π$

Step 1.2.5.2

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

$2 x = \frac{π}{2} + π \cdot \frac{2}{2}$

Step 1.2.5.3

Combine $π$ and $\frac{2}{2}$ .

$2 x = \frac{π}{2} + \frac{π \cdot 2}{2}$

Step 1.2.5.4

Combine the numerators over the common denominator.

$2 x = \frac{π + π \cdot 2}{2}$

Step 1.2.5.5

Simplify the numerator.

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Step 1.2.5.5.1

Move $2$ to the left of $π$ .

$2 x = \frac{π + 2 \cdot π}{2}$

Step 1.2.5.5.2

Add $π$ and $2 π$ .

$2 x = \frac{3 π}{2}$

Step 1.2.6

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ and simplify.

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Step 1.2.6.1

Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{3 π}{2}}{2}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{2}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 1.2.6.3.2

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

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Step 1.2.6.3.2.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 1.2.6.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 1.2.7

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x - π = 2 π - \frac{π}{2}$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Simplify $2 π - \frac{π}{2}$ .

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Step 1.2.8.1.1

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

$2 x - π = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.8.1.2

Combine fractions.

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Step 1.2.8.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$2 x - π = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.8.1.2.2

Combine the numerators over the common denominator.

$2 x - π = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.8.1.3

Simplify the numerator.

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Step 1.2.8.1.3.1

Multiply $2$ by $2$ .

$2 x - π = \frac{4 π - π}{2}$

Step 1.2.8.1.3.2

Subtract $π$ from $4 π$ .

$2 x - π = \frac{3 π}{2}$

Step 1.2.8.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.8.2.1

Add $π$ to both sides of the equation.

$2 x = \frac{3 π}{2} + π$

Step 1.2.8.2.2

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

$2 x = \frac{3 π}{2} + π \cdot \frac{2}{2}$

Step 1.2.8.2.3

Combine $π$ and $\frac{2}{2}$ .

$2 x = \frac{3 π}{2} + \frac{π \cdot 2}{2}$

Step 1.2.8.2.4

Combine the numerators over the common denominator.

$2 x = \frac{3 π + π \cdot 2}{2}$

Step 1.2.8.2.5

Simplify the numerator.

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Step 1.2.8.2.5.1

Move $2$ to the left of $π$ .

$2 x = \frac{3 π + 2 \cdot π}{2}$

Step 1.2.8.2.5.2

Add $3 π$ and $2 π$ .

$2 x = \frac{5 π}{2}$

Step 1.2.8.3

Divide each term in $2 x = \frac{5 π}{2}$ by $2$ and simplify.

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Step 1.2.8.3.1

Divide each term in $2 x = \frac{5 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5 π}{2}}{2}$

Step 1.2.8.3.2

Simplify the left side.

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Step 1.2.8.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.8.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{5 π}{2}}{2}$

Step 1.2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{2}}{2}$

Step 1.2.8.3.3

Simplify the right side.

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Step 1.2.8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{2} \cdot \frac{1}{2}$

Step 1.2.8.3.3.2

Multiply $\frac{5 π}{2} \cdot \frac{1}{2}$ .

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Step 1.2.8.3.3.2.1

Multiply $\frac{5 π}{2}$ by $\frac{1}{2}$ .

$x = \frac{5 π}{2 \cdot 2}$

Step 1.2.8.3.3.2.2

Multiply $2$ by $2$ .

$x = \frac{5 π}{4}$

Step 1.2.9

Find the period of $\cos (2 x - π)$ .

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Step 1.2.9.1

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

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.9.4

Cancel the common factor of $2$ .

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Step 1.2.9.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.9.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.10

The period of the $\cos (2 x - π)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{3 π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 1.2.11

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{4} + \frac{π n}{2}, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4 \cos (2 (0) - π)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cos (2 (0) - π)$

Step 2.2.2

Simplify $4 \cos (2 (0) - π)$ .

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Step 2.2.2.1

Multiply $2$ by $0$ .

$y = 4 \cos (0 - π)$

Step 2.2.2.2

Subtract $π$ from $0$ .

$y = 4 \cos (- π)$

Step 2.2.2.3

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.

$y = 4 (- \cos (0))$

Step 2.2.2.4

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

$y = 4 (- 1 \cdot 1)$

Step 2.2.2.5

Multiply $4 (- 1 \cdot 1)$ .

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Step 2.2.2.5.1

Multiply $- 1$ by $1$ .

$y = 4 \cdot - 1$

Step 2.2.2.5.2

Multiply $4$ by $- 1$ .

$y = - 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 4)$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{4} + \frac{π n}{2}, 0)$ , for any integer $n$

y-intercept(s): $(0, - 4)$

Step 4