Trigonometry Examples

$4 \sec^{2} (x) - 2 = 0$

Step 1

Add $2$ to both sides of the equation.

$4 \sec^{2} (x) = 2$

Step 2

Divide each term in $4 \sec^{2} (x) = 2$ by $4$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $4 \sec^{2} (x) = 2$ by $4$ .

$\frac{4 \sec^{2} (x)}{4} = \frac{2}{4}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{4 \sec^{2} (x)}{4} = \frac{2}{4}$

Step 2.2.1.2

Divide $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \frac{2}{4}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 2.3.1.1

Factor $2$ out of $2$ .

$\sec^{2} (x) = \frac{2 (1)}{4}$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Tap for more steps...

Step 4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 4.2

Any root of $1$ is $1$ .

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

Step 4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4

Combine and simplify the denominator.

Tap for more steps...

Step 4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

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

Step 4.4.6.5

Evaluate the exponent.

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

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.

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

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

Step 7

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

Tap for more steps...

Step 7.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $\frac{\sqrt{2}}{2}$ does not fall in this range, there is no solution.

No solution

Step 8

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

Tap for more steps...

Step 8.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{\sqrt{2}}{2}$ does not fall in this range, there is no solution.

No solution