Trigonometry Examples

$f (x) = 2 \tan (3 x + π)$

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 $x$ .

$0 = 2 \tan (3 x + π)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 \tan (3 x + π) = 0$ .

$2 \tan (3 x + π) = 0$

Step 1.2.2

Divide each term in $2 \tan (3 x + π) = 0$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 \tan (3 x + π) = 0$ by $2$ .

$\frac{2 \tan (3 x + π)}{2} = \frac{0}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 \tan (3 x + π)}{2} = \frac{0}{2}$

Step 1.2.2.2.1.2

Divide $\tan (3 x + π)$ by $1$ .

$\tan (3 x + π) = \frac{0}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $2$ .

$\tan (3 x + π) = 0$

Step 1.2.3

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

$3 x + π = \arctan (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$3 x + π = 0$

Step 1.2.5

Subtract $π$ from both sides of the equation.

$3 x = - π$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{3 x}{3} = \frac{- π}{3}$

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.

$\frac{3 x}{3} = \frac{- π}{3}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- π}{3}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Move the negative in front of the fraction.

$x = - \frac{π}{3}$

Step 1.2.7

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$3 x + π = π + 0$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Add $π$ and $0$ .

$3 x + π = π$

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

Subtract $π$ from both sides of the equation.

$3 x = π - π$

Step 1.2.8.2.2

Subtract $π$ from $π$ .

$3 x = 0$

Step 1.2.8.3

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

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Step 1.2.8.3.1

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

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

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 $3$ .

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Step 1.2.8.3.2.1.1

Cancel the common factor.

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

Step 1.2.8.3.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.8.3.3

Simplify the right side.

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Step 1.2.8.3.3.1

Divide $0$ by $3$ .

$x = 0$

Step 1.2.9

Find the period of $\tan (3 x + π)$ .

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Step 1.2.9.1

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

$\frac{π}{| b |}$

Step 1.2.9.2

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

$\frac{π}{| 3 |}$

Step 1.2.9.3

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

$\frac{π}{3}$

Step 1.2.10

Add $\frac{π}{3}$ to every negative angle to get positive angles.

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Step 1.2.10.1

Add $\frac{π}{3}$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + \frac{π}{3}$

Step 1.2.10.2

Combine the numerators over the common denominator.

$\frac{π - π}{3}$

Step 1.2.10.3

Subtract $π$ from $π$ .

$\frac{0}{3}$

Step 1.2.10.4

Divide $0$ by $3$ .

$0$

Step 1.2.10.5

List the new angles.

$x = 0$

Step 1.2.11

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

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

Step 1.2.12

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 $y$ .

$y = 2 \tan (3 (0) + π)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \tan (3 (0) + π)$

Step 2.2.2

Simplify $2 \tan (3 (0) + π)$ .

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Step 2.2.2.1

Multiply $3$ by $0$ .

$y = 2 \tan (0 + π)$

Step 2.2.2.2

Add $0$ and $π$ .

$y = 2 \tan (π)$

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 tangent is negative in the second quadrant.

$y = 2 (- \tan (0))$

Step 2.2.2.4

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

$y = 2 (- 0)$

Step 2.2.2.5

Multiply $2 (- 0)$ .

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Step 2.2.2.5.1

Multiply $- 1$ by $0$ .

$y = 2 \cdot 0$

Step 2.2.2.5.2

Multiply $2$ by $0$ .

$y = 0$

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