Trigonometry Examples

$g (x) = 5 \cos (3 x) - 5$

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 $x$ .

$0 = 5 \cos (3 x) - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 \cos (3 x) - 5 = 0$ .

$5 \cos (3 x) - 5 = 0$

Step 1.2.2

Add $5$ to both sides of the equation.

$5 \cos (3 x) = 5$

Step 1.2.3

Divide each term in $5 \cos (3 x) = 5$ by $5$ and simplify.

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Step 1.2.3.1

Divide each term in $5 \cos (3 x) = 5$ by $5$ .

$\frac{5 \cos (3 x)}{5} = \frac{5}{5}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $5$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{5 \cos (3 x)}{5} = \frac{5}{5}$

Step 1.2.3.2.1.2

Divide $\cos (3 x)$ by $1$ .

$\cos (3 x) = \frac{5}{5}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $5$ by $5$ .

$\cos (3 x) = 1$

Step 1.2.4

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

$3 x = \arccos (1)$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

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

$3 x = 0$

Step 1.2.6

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.6.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $3$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Divide $0$ by $3$ .

$x = 0$

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.

$3 x = 2 π - 0$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Simplify.

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Step 1.2.8.1.1

Multiply $- 1$ by $0$ .

$3 x = 2 π + 0$

Step 1.2.8.1.2

Add $2 π$ and $0$ .

$3 x = 2 π$

Step 1.2.8.2

Divide each term in $3 x = 2 π$ by $3$ and simplify.

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Step 1.2.8.2.1

Divide each term in $3 x = 2 π$ by $3$ .

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 1.2.8.2.2

Simplify the left side.

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Step 1.2.8.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.8.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2 π}{3}$

Step 1.2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{3}$

Step 1.2.9

Find the period of $\cos (3 x)$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 1.2.10

The period of the $\cos (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

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

Step 1.2.11

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{2 π n}{3}, 0)$ , 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 $y$ .

$y = 5 \cos (3 (0)) - 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 \cos (3 (0)) - 5$

Step 2.2.2

Simplify $5 \cos (3 (0)) - 5$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $3$ by $0$ .

$y = 5 \cos (0) - 5$

Step 2.2.2.1.2

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

$y = 5 \cdot 1 - 5$

Step 2.2.2.1.3

Multiply $5$ by $1$ .

$y = 5 - 5$

Step 2.2.2.2

Subtract $5$ from $5$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(\frac{2 π n}{3}, 0)$ , for any integer $n$

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

Step 4