Trigonometry Examples

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

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

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

$2 u^{2} - u - 1 = 0$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

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 $b = - 1$ .

Tap for more steps...

Step 1.2.1.1

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

$2 u^{2} - (u) - 1 = 0$

Step 1.2.1.2

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

$2 u^{2} + (1 - 2) u - 1 = 0$

Step 1.2.1.3

Apply the distributive property.

$2 u^{2} + 1 u - 2 u - 1 = 0$

Step 1.2.1.4

Multiply $u$ by $1$ .

$2 u^{2} + u - 2 u - 1 = 0$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

$(2 u^{2} + u) - 2 u - 1 = 0$

Step 1.2.2.2

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

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

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

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

Step 1.3

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

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

Step 2

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

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

$\tan (x) - 1 = 0$

Step 3

Set $2 \tan (x) + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $2 \tan (x) + 1$ equal to $0$ .

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

Step 3.2

Solve $2 \tan (x) + 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Subtract $1$ from both sides of the equation.

$2 \tan (x) = - 1$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

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

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

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

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

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

Step 3.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.2.3.1

Move the negative in front of the fraction.

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

Step 3.2.3

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

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

Step 3.2.4

Simplify the right side.

Tap for more steps...

Step 3.2.4.1

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

$x = - 0.4636476$

Step 3.2.5

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.

$x = - 0.4636476 - (3.14159265)$

Step 3.2.6

Simplify the expression to find the second solution.

Tap for more steps...

Step 3.2.6.1

Add $2 π$ to $- 0.4636476 - (3.14159265)$ .

$x = - 0.4636476 - (3.14159265) + 2 π$

Step 3.2.6.2

The resulting angle of $2.67794504$ is positive and coterminal with $- 0.4636476 - (3.14159265)$ .

$x = 2.67794504$

Step 3.2.7

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

Tap for more steps...

Step 3.2.7.1

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

$\frac{π}{| b |}$

Step 3.2.7.2

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

$\frac{π}{| 1 |}$

Step 3.2.7.3

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

$\frac{π}{1}$

Step 3.2.7.4

Divide $π$ by $1$ .

$π$

Step 3.2.8

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

Tap for more steps...

Step 3.2.8.1

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

$- 0.4636476 + π$

Step 3.2.8.2

Replace with decimal approximation.

$3.14159265 - 0.4636476$

Step 3.2.8.3

Subtract $0.4636476$ from $3.14159265$ .

$2.67794504$

Step 3.2.8.4

List the new angles.

$x = 2.67794504$

Step 3.2.9

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

$x = 2.67794504 + π n, 2.67794504 + π n$ , for any integer $n$

Step 4

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

Tap for more steps...

Step 4.1

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

$\tan (x) - 1 = 0$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Add $1$ to 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.

Tap for more steps...

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 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.

$x = π + \frac{π}{4}$

Step 4.2.5

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 4.2.5.1

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

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

Step 4.2.5.2

Combine fractions.

Tap for more steps...

Step 4.2.5.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 4.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 4.2.5.3

Simplify the numerator.

Tap for more steps...

Step 4.2.5.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 4.2.5.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 4.2.6

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

Tap for more steps...

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

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 5

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

$x = 2.67794504 + π n, 2.67794504 + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 6

Consolidate the answers.

Tap for more steps...

Step 6.1

Consolidate $2.67794504 + π n$ and $2.67794504 + π n$ to $2.67794504 + π n$ .

$x = 2.67794504 + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 6.2

Consolidate $\frac{π}{4} + π n$ and $\frac{5 π}{4} + π n$ to $\frac{π}{4} + π n$ .

$x = 2.67794504 + π n, \frac{π}{4} + π n$ , for any integer $n$