Trigonometry Examples

$2 \cos (x) = \sec (x)$

Step 1

Divide each term in $2 \cos (x) = \sec (x)$ by $2 \cos (x)$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $2 \cos (x) = \sec (x)$ by $2 \cos (x)$ .

$\frac{2 \cos (x)}{2 \cos (x)} = \frac{\sec (x)}{2 \cos (x)}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2 \cos (x)} = \frac{\sec (x)}{2 \cos (x)}$

Step 1.2.1.2

Rewrite the expression.

$\frac{\cos (x)}{\cos (x)} = \frac{\sec (x)}{2 \cos (x)}$

Step 1.2.2

Cancel the common factor of $\cos (x)$ .

Tap for more steps...

Step 1.2.2.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} = \frac{\sec (x)}{2 \cos (x)}$

Step 1.2.2.2

Rewrite the expression.

$1 = \frac{\sec (x)}{2 \cos (x)}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Separate fractions.

$1 = \frac{1}{2} \cdot \frac{\sec (x)}{\cos (x)}$

Step 1.3.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$1 = \frac{1}{2} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}$

Step 1.3.3

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$1 = \frac{1}{2} (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})$

Step 1.3.4

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$1 = \frac{1}{2} \cdot \frac{1}{\cos (x) \cos (x)}$

Step 1.3.5

Simplify the denominator.

Tap for more steps...

Step 1.3.5.1

Raise $\cos (x)$ to the power of $1$ .

$1 = \frac{1}{2} \cdot \frac{1}{\cos^{1} (x) \cos (x)}$

Step 1.3.5.2

Raise $\cos (x)$ to the power of $1$ .

$1 = \frac{1}{2} \cdot \frac{1}{\cos^{1} (x) \cos^{1} (x)}$

Step 1.3.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 = \frac{1}{2} \cdot \frac{1}{{\cos (x)}^{1 + 1}}$

Step 1.3.5.4

Add $1$ and $1$ .

$1 = \frac{1}{2} \cdot \frac{1}{\cos^{2} (x)}$

Step 1.3.6

Combine fractions.

Tap for more steps...

Step 1.3.6.1

Combine.

$1 = \frac{1 \cdot 1}{2 \cos^{2} (x)}$

Step 1.3.6.2

Multiply $1$ by $1$ .

$1 = \frac{1}{2 \cos^{2} (x)}$

Step 1.3.7

Multiply by $1$ .

$1 = \frac{1}{2 (\cos^{2} (x) \cdot 1)}$

Step 1.3.8

Separate fractions.

$1 = \frac{1}{2 \cdot (1)} \cdot \frac{1}{\cos^{2} (x)}$

Step 1.3.9

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$1 = \frac{1}{2 \cdot (1)} \sec^{2} (x)$

Step 1.3.10

Multiply $2$ by $1$ .

$1 = \frac{1}{2} \sec^{2} (x)$

Step 1.3.11

Combine $\frac{1}{2}$ and $\sec^{2} (x)$ .

$1 = \frac{\sec^{2} (x)}{2}$

Step 2

Rewrite the equation as $\frac{\sec^{2} (x)}{2} = 1$ .

$\frac{\sec^{2} (x)}{2} = 1$

Step 3

Multiply both sides of the equation by $2$ .

$2 \frac{\sec^{2} (x)}{2} = 2 \cdot 1$

Step 4

Simplify both sides of the equation.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.1.1

Cancel the common factor.

$2 \frac{\sec^{2} (x)}{2} = 2 \cdot 1$

Step 4.1.1.2

Rewrite the expression.

$\sec^{2} (x) = 2 \cdot 1$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Multiply $2$ by $1$ .

$\sec^{2} (x) = 2$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{2}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (x) = \sqrt{2}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (x) = - \sqrt{2}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = \sqrt{2}, - \sqrt{2}$

Step 7

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

$\sec (x) = \sqrt{2}$

$\sec (x) = - \sqrt{2}$

Step 8

Solve for $x$ in $\sec (x) = \sqrt{2}$ .

Tap for more steps...

Step 8.1

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

$x = arcsec (\sqrt{2})$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

The exact value of $arcsec (\sqrt{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 8.3

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.

$x = 2 π - \frac{π}{4}$

Step 8.4

Simplify $2 π - \frac{π}{4}$ .

Tap for more steps...

Step 8.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 8.4.2

Combine fractions.

Tap for more steps...

Step 8.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 8.4.3

Simplify the numerator.

Tap for more steps...

Step 8.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 8.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 8.5

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

Tap for more steps...

Step 8.5.1

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

$\frac{2 π}{| b |}$

Step 8.5.2

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

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 9

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

Tap for more steps...

Step 9.1

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

$x = arcsec (- \sqrt{2})$

Step 9.2

Simplify the right side.

Tap for more steps...

Step 9.2.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 9.3

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.

$x = 2 π - \frac{3 π}{4}$

Step 9.4

Simplify $2 π - \frac{3 π}{4}$ .

Tap for more steps...

Step 9.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 9.4.2

Combine fractions.

Tap for more steps...

Step 9.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 9.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 9.4.3

Simplify the numerator.

Tap for more steps...

Step 9.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 9.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 9.5

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

Tap for more steps...

Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

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

$\frac{2 π}{| 1 |}$

Step 9.5.3

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

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

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

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 10

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 11

Consolidate the answers.

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