Precalculus Examples

$\tan (t) = - 1$ , $\csc (t) = 0$

Step 1

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

$t = \arctan (- 1)$

Step 2

Simplify the right side.

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The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$t = - \frac{π}{4}$

Step 3

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.

$t = - \frac{π}{4} - π$

Step 4

Simplify the expression to find the second solution.

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Add $2 π$ to $- \frac{π}{4} - π$ .

$t = - \frac{π}{4} - π + 2 π$

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$t = \frac{3 π}{4}$

Step 5

Find the period of $\tan (t)$ .

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

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 6

Add $π$ to every negative angle to get positive angles.

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Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Combine fractions.

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Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Simplify the numerator.

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Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

List the new angles.

$t = \frac{3 π}{4}$

Step 7

The period of the $\tan (t)$ function is $π$ so values will repeat every $π$ radians in both directions.

$t = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 8

Consolidate the answers.

$t = \frac{3 π}{4} + π n$ , for any integer $n$

Step 9

Replace all occurrences of $t$ in $\csc (t) = 0$ with $\frac{3 π}{4} + π n$ .

$\csc (\frac{3 π}{4} + π n) = 0$

$t = \frac{3 π}{4} + π n$

Step 10

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 11

Solve the equation for $n$ .

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Rewrite the equation as $\frac{3 π}{4} + π n = t$ .

$\frac{3 π}{4} + π n = t$

No solution

Subtract $\frac{3 π}{4}$ from both sides of the equation.

$π n = t - \frac{3 π}{4}$

No solution

Divide each term in $π n = t - \frac{3 π}{4}$ by $π$ and simplify.

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Divide each term in $π n = t - \frac{3 π}{4}$ by $π$ .

$\frac{π n}{π} = \frac{t}{π} + \frac{- \frac{3 π}{4}}{π}$

No solution

Simplify the left side.

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Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{π n}{π} = \frac{t}{π} + \frac{- \frac{3 π}{4}}{π}$

No solution

Divide $n$ by $1$ .

$n = \frac{t}{π} + \frac{- \frac{3 π}{4}}{π}$

No solution

$n = \frac{t}{π} + \frac{- \frac{3 π}{4}}{π}$

No solution

$n = \frac{t}{π} + \frac{- \frac{3 π}{4}}{π}$

No solution

Simplify the right side.

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

$n = \frac{t}{π} - \frac{3 π}{4} \cdot \frac{1}{π}$

No solution

Cancel the common factor of $π$ .

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Move the leading negative in $- \frac{3 π}{4}$ into the numerator.

$n = \frac{t}{π} + \frac{- 3 π}{4} \cdot \frac{1}{π}$

No solution

Factor $π$ out of $- 3 π$ .

$n = \frac{t}{π} + \frac{π \cdot - 3}{4} \cdot \frac{1}{π}$

No solution

Cancel the common factor.

$n = \frac{t}{π} + \frac{π \cdot - 3}{4} \cdot \frac{1}{π}$

No solution

Rewrite the expression.

$n = \frac{t}{π} + \frac{- 3}{4}$

No solution

$n = \frac{t}{π} + \frac{- 3}{4}$

No solution

Move the negative in front of the fraction.

$n = \frac{t}{π} - \frac{3}{4}$

No solution

$n = \frac{t}{π} - \frac{3}{4}$

No solution

$n = \frac{t}{π} - \frac{3}{4}$

No solution

$n = \frac{t}{π} - \frac{3}{4}$

No solution

Step 12

The simplified system is the arbitrary solution of the original system of equations.

No solution