Precalculus Examples

$\sec^{2} (x) - 2 \tan (x) = 4$

Step 1

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$(1 + \tan^{2} (x)) - 2 \tan (x) = 4$

Step 2

Reorder the polynomial.

$\tan^{2} (x) - 2 \tan (x) + 1 = 4$

Step 3

Substitute $u$ for $\tan (x)$ .

${(u)}^{2} - 2 u + 1 = 4$

Step 4

Subtract $4$ from both sides of the equation.

$u^{2} - 2 u + 1 - 4 = 0$

Step 5

Subtract $4$ from $1$ .

$u^{2} - 2 u - 3 = 0$

Step 6

Factor $u^{2} - 2 u - 3$ using the AC method.

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Step 6.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 6.2

Write the factored form using these integers.

$(u - 3) (u + 1) = 0$

Step 7

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

$u - 3 = 0$

$u + 1 = 0$

Step 8

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 8.2

Add $3$ to both sides of the equation.

$u = 3$

Step 9

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 9.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 10

The final solution is all the values that make $(u - 3) (u + 1) = 0$ true.

$u = 3, - 1$

Step 11

Substitute $\tan (x)$ for $u$ .

$\tan (x) = 3, - 1$

Step 12

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

$\tan (x) = 3$

$\tan (x) = - 1$

Step 13

Solve for $x$ in $\tan (x) = 3$ .

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Step 13.1

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

$x = \arctan (3)$

Step 13.2

Simplify the right side.

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Step 13.2.1

Evaluate $\arctan (3)$ .

$x = 1.24904577$

Step 13.3

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.

$x = (3.14159265) + 1.24904577$

Step 13.4

Solve for $x$ .

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Step 13.4.1

Remove parentheses.

$x = 3.14159265 + 1.24904577$

Step 13.4.2

Remove parentheses.

$x = (3.14159265) + 1.24904577$

Step 13.4.3

Add $3.14159265$ and $1.24904577$ .

$x = 4.39063842$

Step 13.5

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

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Step 13.5.1

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

$\frac{π}{| b |}$

Step 13.5.2

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

$\frac{π}{| 1 |}$

Step 13.5.3

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

$\frac{π}{1}$

Step 13.5.4

Divide $π$ by $1$ .

$π$

Step 13.6

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

$x = 1.24904577 + π n, 4.39063842 + π n$ , for any integer $n$

Step 14

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

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Step 14.1

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

$x = \arctan (- 1)$

Step 14.2

Simplify the right side.

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Step 14.2.1

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

$x = - \frac{π}{4}$

Step 14.3

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.

$x = - \frac{π}{4} - π$

Step 14.4

Simplify the expression to find the second solution.

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Step 14.4.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 14.4.2

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

$x = \frac{3 π}{4}$

Step 14.5

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

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Step 14.5.1

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

$\frac{π}{| b |}$

Step 14.5.2

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

$\frac{π}{| 1 |}$

Step 14.5.3

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

$\frac{π}{1}$

Step 14.5.4

Divide $π$ by $1$ .

$π$

Step 14.6

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

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Step 14.6.1

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

$- \frac{π}{4} + π$

Step 14.6.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 14.6.3

Combine fractions.

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Step 14.6.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 14.6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 14.6.4

Simplify the numerator.

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Step 14.6.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 14.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 14.6.5

List the new angles.

$x = \frac{3 π}{4}$

Step 14.7

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 15

List all of the solutions.

$x = 1.24904577 + π n, 4.39063842 + π n, \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 16

Consolidate the solutions.

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Step 16.1

Consolidate $1.24904577 + π n$ and $4.39063842 + π n$ to $1.24904577 + π n$ .

$x = 1.24904577 + π n, \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 16.2

Consolidate $\frac{3 π}{4} + π n$ and $\frac{3 π}{4} + π n$ to $\frac{3 π}{4} + π n$ .

$x = 1.24904577 + π n, \frac{3 π}{4} + π n$ , for any integer $n$