Trigonometry Examples

$f (x) = 8 \cos^{2} (x) - 4$

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 $x$ .

$0 = 8 \cos^{2} (x) - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $8 \cos^{2} (x) - 4 = 0$ .

$8 \cos^{2} (x) - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$8 \cos^{2} (x) = 4$

Step 1.2.3

Divide each term in $8 \cos^{2} (x) = 4$ by $8$ and simplify.

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Step 1.2.3.1

Divide each term in $8 \cos^{2} (x) = 4$ by $8$ .

$\frac{8 \cos^{2} (x)}{8} = \frac{4}{8}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $8$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{8 \cos^{2} (x)}{8} = \frac{4}{8}$

Step 1.2.3.2.1.2

Divide $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) = \frac{4}{8}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

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Step 1.2.3.3.1.1

Factor $4$ out of $4$ .

$\cos^{2} (x) = \frac{4 (1)}{8}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $4$ out of $8$ .

$\cos^{2} (x) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$\cos^{2} (x) = \frac{4 \cdot 1}{4 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$\cos^{2} (x) = \frac{1}{2}$

Step 1.2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{1}{2}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 1.2.5.2

Any root of $1$ is $1$ .

$\cos (x) = \pm \frac{1}{\sqrt{2}}$

Step 1.2.5.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2.5.4

Combine and simplify the denominator.

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Step 1.2.5.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5.4.5

Add $1$ and $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.5.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.6.4.1

Cancel the common factor.

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.6.4.2

Rewrite the expression.

$\cos (x) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 1.2.5.4.6.5

Evaluate the exponent.

$\cos (x) = \pm \frac{\sqrt{2}}{2}$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 1.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - \frac{\sqrt{2}}{2}$

Step 1.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (x) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 1.2.7

Set up each of the solutions to solve for $x$ .

$\cos (x) = \frac{\sqrt{2}}{2}$

$\cos (x) = - \frac{\sqrt{2}}{2}$

Step 1.2.8

Solve for $x$ in $\cos (x) = \frac{\sqrt{2}}{2}$ .

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Step 1.2.8.1

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 1.2.8.2

Simplify the right side.

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Step 1.2.8.2.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 1.2.8.3

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.

$x = 2 π - \frac{π}{4}$

Step 1.2.8.4

Simplify $2 π - \frac{π}{4}$ .

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Step 1.2.8.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 1.2.8.4.2

Combine fractions.

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Step 1.2.8.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 1.2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 1.2.8.4.3

Simplify the numerator.

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Step 1.2.8.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 1.2.8.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 1.2.8.5

Find the period of $\cos (x)$ .

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Step 1.2.8.5.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.2.8.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.8.6

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 1.2.9

Solve for $x$ in $\cos (x) = - \frac{\sqrt{2}}{2}$ .

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Step 1.2.9.1

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

$x = \arccos (- \frac{\sqrt{2}}{2})$

Step 1.2.9.2

Simplify the right side.

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Step 1.2.9.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 1.2.9.3

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

$x = 2 π - \frac{3 π}{4}$

Step 1.2.9.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 1.2.9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 1.2.9.4.2

Combine fractions.

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Step 1.2.9.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 1.2.9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 1.2.9.4.3

Simplify the numerator.

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Step 1.2.9.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 1.2.9.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 1.2.9.5

Find the period of $\cos (x)$ .

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Step 1.2.9.5.1

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

$\frac{2 π}{| b |}$

Step 1.2.9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.2.9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 1.2.9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.9.6

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 1.2.10

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , 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 $y$ .

$y = 8 \cos^{2} (0) - 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 8 \cos^{2} (0) - 4$

Step 2.2.2

Simplify $8 \cos^{2} (0) - 4$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$y = 8 \cdot 1^{2} - 4$

Step 2.2.2.1.2

One to any power is one.

$y = 8 \cdot 1 - 4$

Step 2.2.2.1.3

Multiply $8$ by $1$ .

$y = 8 - 4$

Step 2.2.2.2

Subtract $4$ from $8$ .

$y = 4$

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