Trigonometry Examples

$y = \cos (3 x)$

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 $x$ .

$0 = \cos (3 x)$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$\cos (3 x) = 0$

Step 1.2.2

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

$3 x = \arccos (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

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

$3 x = \frac{π}{2}$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{3}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{3}$

Step 1.2.4.3.2

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

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Step 1.2.4.3.2.1

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

$x = \frac{π}{2 \cdot 3}$

Step 1.2.4.3.2.2

Multiply $2$ by $3$ .

$x = \frac{π}{6}$

Step 1.2.5

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.

$3 x = 2 π - \frac{π}{2}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Simplify.

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Step 1.2.6.1.1

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

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

Step 1.2.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.6.1.3

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.6.1.4

Multiply $2$ by $2$ .

$3 x = \frac{4 π - π}{2}$

Step 1.2.6.1.5

Subtract $π$ from $4 π$ .

$3 x = \frac{3 π}{2}$

Step 1.2.6.2

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

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Step 1.2.6.2.1

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

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

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

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

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{3}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{3}$

Step 1.2.6.2.3.2

Cancel the common factor of $3$ .

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Step 1.2.6.2.3.2.1

Factor $3$ out of $3 π$ .

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

Step 1.2.6.2.3.2.2

Cancel the common factor.

$x = \frac{3 π}{2} \cdot \frac{1}{3}$

Step 1.2.6.2.3.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{2 π}{| b |}$

Step 1.2.7.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 1.2.7.3

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

$\frac{2 π}{3}$

Step 1.2.8

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.

$x = \frac{π}{6} + \frac{2 π n}{3}, \frac{π}{2} + \frac{2 π n}{3}$ , for any integer $n$

Step 1.2.9

Consolidate the answers.

$x = \frac{π}{6} + \frac{π n}{3}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{6} + \frac{π n}{3}, 0)$ , 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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Multiply $3$ by $0$ .

$y = \cos (0)$

Step 2.2.2.2

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

$y = 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{6} + \frac{π n}{3}, 0)$ , for any integer $n$

y-intercept(s): $(0, 1)$

Step 4