Trigonometry Examples

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

Step 1

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

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

Step 2

Subtract $1$ from $- 1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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 $1$ .

$- 1, 2$

Step 4.2

Write the factored form using these integers.

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

Step 5

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

$u - 1 = 0$

$u + 2 = 0$

Step 6

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

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Step 6.1

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

$u - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 7

Set $u + 2$ equal to $0$ and solve for $u$ .

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Step 7.1

Set $u + 2$ equal to $0$ .

$u + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$u = - 2$

Step 8

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

$u = 1, - 2$

Step 9

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

$\sec (x) = 1, - 2$

Step 10

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

$\sec (x) = 1$

$\sec (x) = - 2$

Step 11

Solve for $x$ in $\sec (x) = 1$ .

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Step 11.1

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

$x = arcsec (1)$

Step 11.2

Simplify the right side.

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Step 11.2.1

The exact value of $arcsec (1)$ is $0$ .

$x = 0$

Step 11.3

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - 0$

Step 11.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 11.5

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

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Step 11.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 11.5.2

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

$\frac{2 π}{| 1 |}$

Step 11.5.3

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

$\frac{2 π}{1}$

Step 11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.6

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 12

Solve for $x$ in $\sec (x) = - 2$ .

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Step 12.1

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

$x = arcsec (- 2)$

Step 12.2

Simplify the right side.

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Step 12.2.1

The exact value of $arcsec (- 2)$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 12.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{2 π}{3}$

Step 12.4

Simplify $2 π - \frac{2 π}{3}$ .

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Step 12.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 12.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 12.5

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

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Step 12.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

The period of the $\sec (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = 2 π n, 2 π + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 14

Consolidate the answers.

$x = \frac{2 π n}{3}$ , for any integer $n$