Trigonometry Examples

$16 \sec^{2} (θ) - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$16 \sec^{2} (θ) = 1$

Step 2

Divide each term in $16 \sec^{2} (θ) = 1$ by $16$ and simplify.

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Step 2.1

Divide each term in $16 \sec^{2} (θ) = 1$ by $16$ .

$\frac{16 \sec^{2} (θ)}{16} = \frac{1}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $16$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 \sec^{2} (θ)}{16} = \frac{1}{16}$

Step 2.2.1.2

Divide $\sec^{2} (θ)$ by $1$ .

$\sec^{2} (θ) = \frac{1}{16}$

Step 3

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

$\sec (θ) = \pm \sqrt{\frac{1}{16}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{16}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{1}{16}}$ as $\frac{\sqrt{1}}{\sqrt{16}}$ .

$\sec (θ) = \pm \frac{\sqrt{1}}{\sqrt{16}}$

Step 4.2

Any root of $1$ is $1$ .

$\sec (θ) = \pm \frac{1}{\sqrt{16}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $16$ as $4^{2}$ .

$\sec (θ) = \pm \frac{1}{\sqrt{4^{2}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\sec (θ) = \pm \frac{1}{4}$

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.

$\sec (θ) = \frac{1}{4}$

Step 5.2

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

$\sec (θ) = - \frac{1}{4}$

Step 5.3

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

$\sec (θ) = \frac{1}{4}, - \frac{1}{4}$

Step 6

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

$\sec (θ) = \frac{1}{4}$

$\sec (θ) = - \frac{1}{4}$

Step 7

Solve for $θ$ in $\sec (θ) = \frac{1}{4}$ .

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Step 7.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $\frac{1}{4}$ does not fall in this range, there is no solution.

No solution

Step 8

Solve for $θ$ in $\sec (θ) = - \frac{1}{4}$ .

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Step 8.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{1}{4}$ does not fall in this range, there is no solution.

No solution