Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

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

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

Step 1.3.4.4

Add $1$ and $1$ .

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

Step 1.3.5

Simplify with factoring out.

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Step 1.3.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.3.5.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 1.3.6

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

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

Step 2

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

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

Step 3

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

$\sec (x) = \pm \sqrt{1}$

Step 4

Any root of $1$ is $1$ .

$\sec (x) = \pm 1$

Step 5

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

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Step 5.1

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

$\sec (x) = 1$

Step 5.2

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

$\sec (x) = - 1$

Step 5.3

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

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

Step 6

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

$\sec (x) = 1$

$\sec (x) = - 1$

Step 7

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

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Step 7.1

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

$x = arcsec (1)$

Step 7.2

Simplify the right side.

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Step 7.2.1

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

$x = 0$

Step 7.3

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 7.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 7.5

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

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Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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 8

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

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Step 8.1

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

$x = arcsec (- 1)$

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $arcsec (- 1)$ is $π$ .

$x = π$

Step 8.3

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.

$x = 2 π - π$

Step 8.4

Subtract $π$ from $2 π$ .

$x = π$

Step 8.5

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

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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.

$x = π + 2 π n$ , for any integer $n$

Step 9

List all of the solutions.

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

Step 10

Consolidate the answers.

$x = π n$ , for any integer $n$