Trigonometry Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

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

$u^{2} - u - 2 = 0$

Step 1.2

Factor $u^{2} - u - 2$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2.2

Write the factored form using these integers.

$(u - 2) (u + 1) = 0$

Step 1.3

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

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

Step 2

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

$\tan (x) - 2 = 0$

$\tan (x) + 1 = 0$

Step 3

Set $\tan (x) - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\tan (x) - 2$ equal to $0$ .

$\tan (x) - 2 = 0$

Step 3.2

Solve $\tan (x) - 2 = 0$ for $x$ .

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Step 3.2.1

Add $2$ to both sides of the equation.

$\tan (x) = 2$

Step 3.2.2

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

$x = \arctan (2)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate $\arctan (2)$ .

$x = 1.10714871$

Step 3.2.4

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.

$x = (3.14159265) + 1.10714871$

Step 3.2.5

Solve for $x$ .

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Step 3.2.5.1

Remove parentheses.

$x = 3.14159265 + 1.10714871$

Step 3.2.5.2

Remove parentheses.

$x = (3.14159265) + 1.10714871$

Step 3.2.5.3

Add $3.14159265$ and $1.10714871$ .

$x = 4.24874137$

Step 3.2.6

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

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Step 3.2.6.1

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

$\frac{π}{| b |}$

Step 3.2.6.2

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

$\frac{π}{| 1 |}$

Step 3.2.6.3

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

$\frac{π}{1}$

Step 3.2.6.4

Divide $π$ by $1$ .

$π$

Step 3.2.7

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

$x = 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 4

Set $\tan (x) + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\tan (x) + 1$ equal to $0$ .

$\tan (x) + 1 = 0$

Step 4.2

Solve $\tan (x) + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 4.2.2

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

$x = \arctan (- 1)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x = - \frac{π}{4}$

Step 4.2.4

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.

$x = - \frac{π}{4} - π$

Step 4.2.5

Simplify the expression to find the second solution.

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Step 4.2.5.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 4.2.5.2

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

$x = \frac{3 π}{4}$

Step 4.2.6

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

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Step 4.2.6.1

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

$\frac{π}{| b |}$

Step 4.2.6.2

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

$\frac{π}{| 1 |}$

Step 4.2.6.3

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

$\frac{π}{1}$

Step 4.2.6.4

Divide $π$ by $1$ .

$π$

Step 4.2.7

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

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Step 4.2.7.1

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

$- \frac{π}{4} + π$

Step 4.2.7.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 4.2.7.3

Combine fractions.

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Step 4.2.7.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 4.2.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 4.2.7.4

Simplify the numerator.

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Step 4.2.7.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 4.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 4.2.7.5

List the new angles.

$x = \frac{3 π}{4}$

Step 4.2.8

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

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

Step 5

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

$x = 1.10714871 + π n, 4.24874137 + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 6

Consolidate $1.10714871 + π n$ and $4.24874137 + π n$ to $1.10714871 + π n$ .

$x = 1.10714871 + π n, \frac{3 π}{4} + π n$ , for any integer $n$