Solve for t tan(3t+pi/2)^2=1/3

$\tan^{2} (3 t + \frac{π}{2}) = \frac{1}{3}$

Step 1

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

$\tan (3 t + \frac{π}{2}) = \pm \sqrt{\frac{1}{3}}$

Step 2

Simplify $\pm \sqrt{\frac{1}{3}}$ .

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Any root of $1$ is $1$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{3^{1}}$

Evaluate the exponent.

$\tan (3 t + \frac{π}{2}) = \pm \frac{\sqrt{3}}{3}$

Step 3

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

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

$\tan (3 t + \frac{π}{2}) = \frac{\sqrt{3}}{3}$

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

$\tan (3 t + \frac{π}{2}) = - \frac{\sqrt{3}}{3}$

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

$\tan (3 t + \frac{π}{2}) = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

Step 4

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

$\tan (3 t + \frac{π}{2}) = \frac{\sqrt{3}}{3}$

$\tan (3 t + \frac{π}{2}) = - \frac{\sqrt{3}}{3}$

Step 5

Solve for $t$ in $\tan (3 t + \frac{π}{2}) = \frac{\sqrt{3}}{3}$ .

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

$3 t + \frac{π}{2} = \arctan (\frac{\sqrt{3}}{3})$

Simplify the right side.

The exact value of $\arctan (\frac{\sqrt{3}}{3})$ is $\frac{π}{6}$ .

$3 t + \frac{π}{2} = \frac{π}{6}$

Move all terms not containing $t$ to the right side of the equation.

Subtract $\frac{π}{2}$ from both sides of the equation.

$3 t = \frac{π}{6} - \frac{π}{2}$

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 t = \frac{π}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$3 t = \frac{π}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$3 t = \frac{π}{6} - \frac{π \cdot 3}{6}$

Combine the numerators over the common denominator.

$3 t = \frac{π - π \cdot 3}{6}$

Simplify the numerator.

Multiply $3$ by $- 1$ .

$3 t = \frac{π - 3 π}{6}$

Subtract $3 π$ from $π$ .

$3 t = \frac{- 2 π}{6}$

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2 π$ .

$3 t = \frac{2 (- π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$3 t = \frac{2 (- π)}{2 (3)}$

Cancel the common factor.

$3 t = \frac{2 (- π)}{2 \cdot 3}$

Rewrite the expression.

$3 t = \frac{- π}{3}$

Move the negative in front of the fraction.

$3 t = - \frac{π}{3}$

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

Divide each term in $3 t = - \frac{π}{3}$ by $3$ .

$\frac{3 t}{3} = \frac{- \frac{π}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} = \frac{- \frac{π}{3}}{3}$

Divide $t$ by $1$ .

$t = \frac{- \frac{π}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = - \frac{π}{3} \cdot \frac{1}{3}$

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

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

$t = - \frac{π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$t = - \frac{π}{9}$

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.

$3 t + \frac{π}{2} = π + \frac{π}{6}$

Solve for $t$ .

Simplify $π + \frac{π}{6}$ .

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$3 t + \frac{π}{2} = π \cdot \frac{6}{6} + \frac{π}{6}$

Combine fractions.

Combine $π$ and $\frac{6}{6}$ .

$3 t + \frac{π}{2} = \frac{π \cdot 6}{6} + \frac{π}{6}$

Combine the numerators over the common denominator.

$3 t + \frac{π}{2} = \frac{π \cdot 6 + π}{6}$

Simplify the numerator.

Move $6$ to the left of $π$ .

$3 t + \frac{π}{2} = \frac{6 \cdot π + π}{6}$

Add $6 π$ and $π$ .

$3 t + \frac{π}{2} = \frac{7 π}{6}$

Move all terms not containing $t$ to the right side of the equation.

Subtract $\frac{π}{2}$ from both sides of the equation.

$3 t = \frac{7 π}{6} - \frac{π}{2}$

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 t = \frac{7 π}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$3 t = \frac{7 π}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$3 t = \frac{7 π}{6} - \frac{π \cdot 3}{6}$

Combine the numerators over the common denominator.

$3 t = \frac{7 π - π \cdot 3}{6}$

Simplify the numerator.

Multiply $3$ by $- 1$ .

$3 t = \frac{7 π - 3 π}{6}$

Subtract $3 π$ from $7 π$ .

$3 t = \frac{4 π}{6}$

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4 π$ .

$3 t = \frac{2 (2 π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$3 t = \frac{2 (2 π)}{2 (3)}$

Cancel the common factor.

$3 t = \frac{2 (2 π)}{2 \cdot 3}$

Rewrite the expression.

$3 t = \frac{2 π}{3}$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $t$ by $1$ .

$t = \frac{\frac{2 π}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{2 π}{3} \cdot \frac{1}{3}$

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

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

$t = \frac{2 π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$t = \frac{2 π}{9}$

Find the period of $\tan (3 t + \frac{π}{2})$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 3 |}$

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

$\frac{π}{3}$

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

Add $\frac{π}{3}$ to $- \frac{π}{9}$ to find the positive angle.

$- \frac{π}{9} + \frac{π}{3}$

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π}{3} \cdot \frac{3}{3} - \frac{π}{9}$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{π \cdot 3}{3 \cdot 3} - \frac{π}{9}$

Multiply $3$ by $3$ .

$\frac{π \cdot 3}{9} - \frac{π}{9}$

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - π}{9}$

Simplify the numerator.

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - π}{9}$

Subtract $π$ from $3 π$ .

$\frac{2 π}{9}$

List the new angles.

$t = \frac{2 π}{9}$

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

$t = \frac{2 π}{9} + \frac{π n}{3}, \frac{2 π}{9} + \frac{π n}{3}$ , for any integer $n$

Step 6

Solve for $t$ in $\tan (3 t + \frac{π}{2}) = - \frac{\sqrt{3}}{3}$ .

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

$3 t + \frac{π}{2} = \arctan (- \frac{\sqrt{3}}{3})$

Simplify the right side.

The exact value of $\arctan (- \frac{\sqrt{3}}{3})$ is $- \frac{π}{6}$ .

$3 t + \frac{π}{2} = - \frac{π}{6}$

Move all terms not containing $t$ to the right side of the equation.

Subtract $\frac{π}{2}$ from both sides of the equation.

$3 t = - \frac{π}{6} - \frac{π}{2}$

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 t = - \frac{π}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$3 t = - \frac{π}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$3 t = - \frac{π}{6} - \frac{π \cdot 3}{6}$

Combine the numerators over the common denominator.

$3 t = \frac{- π - π \cdot 3}{6}$

Simplify the numerator.

Multiply $3$ by $- 1$ .

$3 t = \frac{- π - 3 π}{6}$

Subtract $3 π$ from $- π$ .

$3 t = \frac{- 4 π}{6}$

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4 π$ .

$3 t = \frac{2 (- 2 π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$3 t = \frac{2 (- 2 π)}{2 (3)}$

Cancel the common factor.

$3 t = \frac{2 (- 2 π)}{2 \cdot 3}$

Rewrite the expression.

$3 t = \frac{- 2 π}{3}$

Move the negative in front of the fraction.

$3 t = - \frac{2 π}{3}$

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

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

$\frac{3 t}{3} = \frac{- \frac{2 π}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} = \frac{- \frac{2 π}{3}}{3}$

Divide $t$ by $1$ .

$t = \frac{- \frac{2 π}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = - \frac{2 π}{3} \cdot \frac{1}{3}$

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

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

$t = - \frac{2 π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$t = - \frac{2 π}{9}$

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

$3 t + \frac{π}{2} = - \frac{π}{6} - π$

Simplify the expression to find the second solution.

Add $2 π$ to $- \frac{π}{6} - π$ .

$3 t + \frac{π}{2} = - \frac{π}{6} - π + 2 π$

The resulting angle of $\frac{5 π}{6}$ is positive and coterminal with $- \frac{π}{6} - π$ .

$3 t + \frac{π}{2} = \frac{5 π}{6}$

Solve for $t$ .

Move all terms not containing $t$ to the right side of the equation.

Subtract $\frac{π}{2}$ from both sides of the equation.

$3 t = \frac{5 π}{6} - \frac{π}{2}$

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 t = \frac{5 π}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

$3 t = \frac{5 π}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$3 t = \frac{5 π}{6} - \frac{π \cdot 3}{6}$

Combine the numerators over the common denominator.

$3 t = \frac{5 π - π \cdot 3}{6}$

Simplify the numerator.

Multiply $3$ by $- 1$ .

$3 t = \frac{5 π - 3 π}{6}$

Subtract $3 π$ from $5 π$ .

$3 t = \frac{2 π}{6}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 π$ .

$3 t = \frac{2 (π)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$3 t = \frac{2 π}{2 \cdot 3}$

Cancel the common factor.

$3 t = \frac{2 π}{2 \cdot 3}$

Rewrite the expression.

$3 t = \frac{π}{3}$

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

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

$\frac{3 t}{3} = \frac{\frac{π}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 t}{3} = \frac{\frac{π}{3}}{3}$

Divide $t$ by $1$ .

$t = \frac{\frac{π}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{π}{3} \cdot \frac{1}{3}$

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

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

$t = \frac{π}{3 \cdot 3}$

Multiply $3$ by $3$ .

$t = \frac{π}{9}$

Find the period of $\tan (3 t + \frac{π}{2})$ .

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

$\frac{π}{| b |}$

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

$\frac{π}{| 3 |}$

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

$\frac{π}{3}$

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

Add $\frac{π}{3}$ to $- \frac{2 π}{9}$ to find the positive angle.

$- \frac{2 π}{9} + \frac{π}{3}$

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{π}{3} \cdot \frac{3}{3} - \frac{2 π}{9}$

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{π \cdot 3}{3 \cdot 3} - \frac{2 π}{9}$

Multiply $3$ by $3$ .

$\frac{π \cdot 3}{9} - \frac{2 π}{9}$

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - 2 π}{9}$

Simplify the numerator.

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - 2 π}{9}$

Subtract $2 π$ from $3 π$ .

$\frac{π}{9}$

List the new angles.

$t = \frac{π}{9}$

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

$t = \frac{π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}$ , for any integer $n$

Step 7

List all of the solutions.

$t = \frac{2 π}{9} + \frac{π n}{3}, \frac{2 π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}$ , for any integer $n$

Step 8

Consolidate the solutions.

Consolidate $\frac{2 π}{9} + \frac{π n}{3}$ and $\frac{2 π}{9} + \frac{π n}{3}$ to $\frac{2 π}{9} + \frac{π n}{3}$ .

$t = \frac{2 π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}$ , for any integer $n$

Consolidate $\frac{π}{9} + \frac{π n}{3}$ and $\frac{π}{9} + \frac{π n}{3}$ to $\frac{π}{9} + \frac{π n}{3}$ .

$t = \frac{2 π}{9} + \frac{π n}{3}, \frac{π}{9} + \frac{π n}{3}$ , for any integer $n$

Solve for t tan(3t+pi/2)^2=1/3

Related Questions

Support LCMGCF.com by disabling your adblocker.