Trigonometry Examples

$16 \cos^{2} (θ) - 25 = 0$

Step 1

Add $25$ to both sides of the equation.

$16 \cos^{2} (θ) = 25$

Step 2

Divide each term in $16 \cos^{2} (θ) = 25$ by $16$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $16 \cos^{2} (θ) = 25$ by $16$ .

$\frac{16 \cos^{2} (θ)}{16} = \frac{25}{16}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $16$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{16 \cos^{2} (θ)}{16} = \frac{25}{16}$

Step 2.2.1.2

Divide $\cos^{2} (θ)$ by $1$ .

$\cos^{2} (θ) = \frac{25}{16}$

Step 3

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

$\cos (θ) = \pm \sqrt{\frac{25}{16}}$

Step 4

Simplify $\pm \sqrt{\frac{25}{16}}$ .

Tap for more steps...

Step 4.1

Rewrite $\sqrt{\frac{25}{16}}$ as $\frac{\sqrt{25}}{\sqrt{16}}$ .

$\cos (θ) = \pm \frac{\sqrt{25}}{\sqrt{16}}$

Step 4.2

Simplify the numerator.

Tap for more steps...

Step 4.2.1

Rewrite $25$ as $5^{2}$ .

$\cos (θ) = \pm \frac{\sqrt{5^{2}}}{\sqrt{16}}$

Step 4.2.2

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

$\cos (θ) = \pm \frac{5}{\sqrt{16}}$

Step 4.3

Simplify the denominator.

Tap for more steps...

Step 4.3.1

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

$\cos (θ) = \pm \frac{5}{\sqrt{4^{2}}}$

Step 4.3.2

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

$\cos (θ) = \pm \frac{5}{4}$

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.

$\cos (θ) = \frac{5}{4}$

Step 5.2

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

$\cos (θ) = - \frac{5}{4}$

Step 5.3

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

$\cos (θ) = \frac{5}{4}, - \frac{5}{4}$

Step 6

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

$\cos (θ) = \frac{5}{4}$

$\cos (θ) = - \frac{5}{4}$

Step 7

Solve for $θ$ in $\cos (θ) = \frac{5}{4}$ .

Tap for more steps...

Step 7.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $\frac{5}{4}$ does not fall in this range, there is no solution.

No solution

Step 8

Solve for $θ$ in $\cos (θ) = - \frac{5}{4}$ .

Tap for more steps...

Step 8.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{5}{4}$ does not fall in this range, there is no solution.

No solution