Precalculus Examples

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

Step 1

Divide each term in $\sec (\frac{θ}{2}) = \cos (\frac{θ}{2})$ by $\sec (\frac{θ}{2})$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $\sec (\frac{θ}{2}) = \cos (\frac{θ}{2})$ by $\sec (\frac{θ}{2})$ .

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

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $\sec (\frac{θ}{2})$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Rewrite $\sec (\frac{θ}{2})$ in terms of sines and cosines.

$1 = \frac{\cos (\frac{θ}{2})}{\frac{1}{\cos (\frac{θ}{2})}}$

Step 1.3.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{θ}{2})}$ .

$1 = \cos (\frac{θ}{2}) \cos (\frac{θ}{2})$

Step 1.3.3

Simplify.

Tap for more steps...

Step 1.3.3.1

Raise $\cos (\frac{θ}{2})$ to the power of $1$ .

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

Step 1.3.3.2

Raise $\cos (\frac{θ}{2})$ to the power of $1$ .

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

Step 1.3.3.3

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

$1 = {\cos (\frac{θ}{2})}^{1 + 1}$

Step 1.3.3.4

Add $1$ and $1$ .

$1 = \cos^{2} (\frac{θ}{2})$

Step 2

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

$\cos^{2} (\frac{θ}{2}) = 1$

Step 3

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

$\cos (\frac{θ}{2}) = \pm \sqrt{1}$

Step 4

Any root of $1$ is $1$ .

$\cos (\frac{θ}{2}) = \pm 1$

Step 5

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

Tap for more steps...

Step 5.1

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

$\cos (\frac{θ}{2}) = 1$

Step 5.2

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

$\cos (\frac{θ}{2}) = - 1$

Step 5.3

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

$\cos (\frac{θ}{2}) = 1, - 1$

Step 6

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

$\cos (\frac{θ}{2}) = 1$

$\cos (\frac{θ}{2}) = - 1$

Step 7

Solve for $θ$ in $\cos (\frac{θ}{2}) = 1$ .

Tap for more steps...

Step 7.1

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$\frac{θ}{2} = \arccos (1)$

Step 7.2

Simplify the right side.

Tap for more steps...

Step 7.2.1

The exact value of $\arccos (1)$ is $0$ .

$\frac{θ}{2} = 0$

Step 7.3

Set the numerator equal to zero.

$θ = 0$

Step 7.4

The cosine 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.

$\frac{θ}{2} = 2 π - 0$

Step 7.5

Solve for $θ$ .

Tap for more steps...

Step 7.5.1

Multiply both sides of the equation by $2$ .

$2 \frac{θ}{2} = 2 (2 π - 0)$

Step 7.5.2

Simplify both sides of the equation.

Tap for more steps...

Step 7.5.2.1

Simplify the left side.

Tap for more steps...

Step 7.5.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.5.2.1.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 (2 π - 0)$

Step 7.5.2.1.1.2

Rewrite the expression.

$θ = 2 (2 π - 0)$

Step 7.5.2.2

Simplify the right side.

Tap for more steps...

Step 7.5.2.2.1

Simplify $2 (2 π - 0)$ .

Tap for more steps...

Step 7.5.2.2.1.1

Subtract $0$ from $2 π$ .

$θ = 2 (2 π)$

Step 7.5.2.2.1.2

Multiply $2$ by $2$ .

$θ = 4 π$

Step 7.6

Find the period of $\cos (\frac{θ}{2})$ .

Tap for more steps...

Step 7.6.1

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

$\frac{2 π}{| b |}$

Step 7.6.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 7.6.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 7.6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 7.6.5

Multiply $2$ by $2$ .

$4 π$

Step 7.7

The period of the $\cos (\frac{θ}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$θ = 4 π n, 4 π + 4 π n$ , for any integer $n$

Step 8

Solve for $θ$ in $\cos (\frac{θ}{2}) = - 1$ .

Tap for more steps...

Step 8.1

Take the inverse cosine of both sides of the equation to extract $θ$ from inside the cosine.

$\frac{θ}{2} = \arccos (- 1)$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

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

$\frac{θ}{2} = π$

Step 8.3

Multiply both sides of the equation by $2$ .

$2 \frac{θ}{2} = 2 π$

Step 8.4

Simplify the left side.

Tap for more steps...

Step 8.4.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.4.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 π$

Step 8.4.1.2

Rewrite the expression.

$θ = 2 π$

Step 8.5

The cosine 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.

$\frac{θ}{2} = 2 π - π$

Step 8.6

Solve for $θ$ .

Tap for more steps...

Step 8.6.1

Multiply both sides of the equation by $2$ .

$2 \frac{θ}{2} = 2 (2 π - π)$

Step 8.6.2

Simplify both sides of the equation.

Tap for more steps...

Step 8.6.2.1

Simplify the left side.

Tap for more steps...

Step 8.6.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.6.2.1.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 (2 π - π)$

Step 8.6.2.1.1.2

Rewrite the expression.

$θ = 2 (2 π - π)$

Step 8.6.2.2

Simplify the right side.

Tap for more steps...

Step 8.6.2.2.1

Subtract $π$ from $2 π$ .

$θ = 2 π$

Step 8.7

Find the period of $\cos (\frac{θ}{2})$ .

Tap for more steps...

Step 8.7.1

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

$\frac{2 π}{| b |}$

Step 8.7.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 8.7.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 8.7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 8.7.5

Multiply $2$ by $2$ .

$4 π$

Step 8.8

The period of the $\cos (\frac{θ}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$θ = 2 π + 4 π n$ , for any integer $n$

Step 9

List all of the solutions.

$θ = 4 π n, 4 π + 4 π n, 2 π + 4 π n$ , for any integer $n$

Step 10

Consolidate the answers.

$θ = 2 π n$ , for any integer $n$