Trigonometry Examples

$y = \sin (3 x)$

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 $x$ .

$0 = \sin (3 x)$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$\sin (3 x) = 0$

Step 1.2.2

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

$3 x = \arcsin (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

The exact value of $\arcsin (0)$ is $0$ .

$3 x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

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

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.

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

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $0$ by $3$ .

$x = 0$

Step 1.2.5

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

$3 x = π - 0$

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

Multiply $- 1$ by $0$ .

$3 x = π + 0$

Step 1.2.6.1.2

Add $π$ and $0$ .

$3 x = π$

Step 1.2.6.2

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

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Step 1.2.6.2.1

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

$\frac{3 x}{3} = \frac{π}{3}$

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.

$\frac{3 x}{3} = \frac{π}{3}$

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Step 1.2.7

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

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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 $\sin (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

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

Step 1.2.9

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Multiply $3$ by $0$ .

$y = \sin (0)$

Step 2.2.2.2

The exact value of $\sin (0)$ is $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4