Trigonometry Examples

$2 \tan^{2} (x) - \tan (x) - 1 = 0$

Factor the left side of the equation.

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Let $u = \tan (x)$ . Substitute $u$ for all occurrences of $\tan (x)$ .

$2 u^{2} - u - 1$

Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Factor $- 1$ out of $- u$ .

$2 u^{2} - (u) - 1$

Rewrite $- 1$ as $- 2$ plus $1$

$2 u^{2} + (- 2 + 1) u - 1$

Apply the distributive property.

$2 u^{2} + (- 2 u + 1 u) - 1$

Multiply $u$ by $1$ .

$2 u^{2} + (- 2 u + u) - 1$

Remove parentheses.

$2 u^{2} - 2 u + u - 1$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(2 u^{2} - 2 u) + (u - 1)$

Factor out the greatest common factor (GCF) from each group.

$2 u (u - 1) + 1 (u - 1)$

Factor the polynomial by factoring out the greatest common factor, $u - 1$ .

$(u - 1) (2 u + 1)$

Replace all occurrences of $u$ with $\tan (x)$ .

$(\tan (x) - 1) (2 \tan (x) + 1)$

Replace the left side with the factored expression.

$(\tan (x) - 1) (2 \tan (x) + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan (x) - 1 = 0$

$2 \tan (x) + 1 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$\tan (x) - 1 = 0$

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Simplify the expression to find the first solution.

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

$x = \arctan (1)$

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

$x = \frac{π}{4}$

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

$x = π + \frac{π}{4}$

Simplify the expression to find the second solution.

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{π}{1} \cdot \frac{4}{4} + \frac{π}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{π \cdot 4}{1 \cdot 4} + \frac{π}{4}$

Multiply $4$ by $1$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Simplify the numerator.

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Move $4$ to the left of the expression $π \cdot 4$ .

$x = \frac{4 \cdot π + π}{4}$

Multiply $4$ by $π$ .

$x = \frac{4 π + π}{4}$

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Find the period.

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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.

$\frac{π}{| 1 |}$

Solve the equation.

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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 $1$ .

$π$

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

$x = \frac{π}{4} \pm π n, \frac{5 π}{4} \pm π n$

Consolidate the answers.

$x = \frac{π}{4} \pm π n$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$2 \tan (x) + 1 = 0$

Subtract $1$ from both sides of the equation.

$2 \tan (x) = - 1$

Divide each term by $2$ and simplify.

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Divide each term in $2 \tan (x) = - 1$ by $2$ .

$\frac{2 \tan (x)}{2} = - \frac{1}{2}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{2 \tan (x)}{2} = - \frac{1}{2}$

Divide $\tan (x)$ by $1$ .

$\tan (x) = - \frac{1}{2}$

Simplify the expression to find the first solution.

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

$x = \arctan (- \frac{1}{2})$

Evaluate $\arctan (- \frac{1}{2})$ .

$x = - 0.4636476$

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.

$x = - 0.4636476 - (3.14159265)$

Simplify the expression to find the second solution.

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Multiply $- 1$ by $3.14159265$ .

$x = - 0.4636476 - 3.14159265$

Subtract $3.14159265$ from $- 0.4636476$ .

$x = - 3.60524026$

Add $360 °$ to $- 3.60524026 °$ .

$x = - 3.60524026 ° + 360 °$

The resulting angle of $356.39475973 °$ is positive and coterminal with $- 3.60524026$ .

$x = 356.39475973 °$

Find the period.

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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.

$\frac{π}{| 1 |}$

Solve the equation.

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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 $1$ .

$π$

Add $2 π$ to every negative angle to get positive angles.

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Add $2 π$ to $- 0.4636476$ to find the positive angle.

$- 0.4636476 + 2 π$

Subtract $0.4636476$ from $2 π$ .

$5.81953769$

List the new angles.

$x = 5.81953769$

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

$x = 5.81953769 \pm π n, 356.39475973 \pm π n$

The final solution is all the values that make $(\tan (x) - 1) (2 \tan (x) + 1) = 0$ true.

$x = \frac{π}{4} \pm π n, 5.81953769 \pm π n, 356.39475973 \pm π n$

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