Trigonometry Examples

$2 \sec^{2} (x) = 3 - \tan (x)$

Step 1

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

$2 (1 + \tan^{2} (x)) = 3 - \tan (x)$

Step 2

Apply the distributive property.

$2 \cdot 1 + 2 \tan^{2} (x) = 3 - \tan (x)$

Step 3

Multiply $2$ by $1$ .

$2 + 2 \tan^{2} (x) = 3 - \tan (x)$

Step 4

Reorder the polynomial.

$2 \tan^{2} (x) + 2 = 3 - \tan (x)$

Step 5

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

$2 {(u)}^{2} + 2 = 3 - (u)$

Step 6

Add $u$ to both sides of the equation.

$2 u^{2} + 2 + u = 3$

Step 7

Subtract $3$ from both sides of the equation.

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

Step 8

Subtract $3$ from $2$ .

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

Step 9

Factor by grouping.

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Step 9.1

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 $b = 1$ .

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Step 9.1.1

Multiply by $1$ .

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

Step 9.1.2

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

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

Step 9.1.3

Apply the distributive property.

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

Step 9.2

Factor out the greatest common factor from each group.

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Step 9.2.1

Group the first two terms and the last two terms.

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

Step 9.2.2

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

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

Step 9.3

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

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

Step 10

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

$2 u - 1 = 0$

$u + 1 = 0$

Step 11

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 11.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 11.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 11.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 11.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 11.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $2$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 11.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 12

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

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Step 12.1

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

$u + 1 = 0$

Step 12.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 13

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

$u = \frac{1}{2}, - 1$

Step 14

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

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

Step 15

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

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

$\tan (x) = - 1$

Step 16

Solve for $x$ in $\tan (x) = \frac{1}{2}$ .

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Step 16.1

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

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

Step 16.2

Simplify the right side.

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Step 16.2.1

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

$x = 0.4636476$

Step 16.3

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.

$x = (3.14159265) + 0.4636476$

Step 16.4

Solve for $x$ .

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Step 16.4.1

Remove parentheses.

$x = 3.14159265 + 0.4636476$

Step 16.4.2

Remove parentheses.

$x = (3.14159265) + 0.4636476$

Step 16.4.3

Add $3.14159265$ and $0.4636476$ .

$x = 3.60524026$

Step 16.5

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

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Step 16.5.1

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

$\frac{π}{| b |}$

Step 16.5.2

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

$\frac{π}{| 1 |}$

Step 16.5.3

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

$\frac{π}{1}$

Step 16.5.4

Divide $π$ by $1$ .

$π$

Step 16.6

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

$x = 0.4636476 + π n, 3.60524026 + π n$ , for any integer $n$

Step 17

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

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Step 17.1

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

$x = \arctan (- 1)$

Step 17.2

Simplify the right side.

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Step 17.2.1

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

$x = - \frac{π}{4}$

Step 17.3

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 17.4

Simplify the expression to find the second solution.

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Step 17.4.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 17.4.2

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

$x = \frac{3 π}{4}$

Step 17.5

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

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Step 17.5.1

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

$\frac{π}{| b |}$

Step 17.5.2

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

$\frac{π}{| 1 |}$

Step 17.5.3

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

$\frac{π}{1}$

Step 17.5.4

Divide $π$ by $1$ .

$π$

Step 17.6

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

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Step 17.6.1

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

$- \frac{π}{4} + π$

Step 17.6.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 17.6.3

Combine fractions.

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Step 17.6.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 17.6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 17.6.4

Simplify the numerator.

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Step 17.6.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 17.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 17.6.5

List the new angles.

$x = \frac{3 π}{4}$

Step 17.7

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 18

List all of the solutions.

$x = 0.4636476 + π n, 3.60524026 + π n, \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 19

Consolidate the solutions.

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Step 19.1

Consolidate $0.4636476 + π n$ and $3.60524026 + π n$ to $0.4636476 + π n$ .

$x = 0.4636476 + π n, \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 19.2

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

$x = 0.4636476 + π n, \frac{3 π}{4} + π n$ , for any integer $n$