Trigonometry Examples

$\cos (θ) = \frac{21}{\sqrt{768}}$

Step 1

Simplify $\frac{21}{\sqrt{768}}$ .

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

Rewrite $768$ as $16^{2} \cdot 3$ .

Tap for more steps...

Step 1.1.1.1

Factor $256$ out of $768$ .

$\cos (θ) = \frac{21}{\sqrt{256 (3)}}$

Step 1.1.1.2

Rewrite $256$ as $16^{2}$ .

$\cos (θ) = \frac{21}{\sqrt{16^{2} \cdot 3}}$

Step 1.1.2

Pull terms out from under the radical.

$\cos (θ) = \frac{21}{16 \sqrt{3}}$

Step 1.2

Multiply $\frac{21}{16 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (θ) = \frac{21}{16 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.3

Combine and simplify the denominator.

Tap for more steps...

Step 1.3.1

Multiply $\frac{21}{16 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 \sqrt{3} \sqrt{3}}$

Step 1.3.2

Move $\sqrt{3}$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 (\sqrt{3} \sqrt{3})}$

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 1.3.4

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 1.3.5

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

$\cos (θ) = \frac{21 \sqrt{3}}{16 {\sqrt{3}}^{1 + 1}}$

Step 1.3.6

Add $1$ and $1$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 {\sqrt{3}}^{2}}$

Step 1.3.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cos (θ) = \frac{21 \sqrt{3}}{16 {(3^{\frac{1}{2}})}^{2}}$

Step 1.3.7.2

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

$\cos (θ) = \frac{21 \sqrt{3}}{16 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 1.3.7.3

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

$\cos (θ) = \frac{21 \sqrt{3}}{16 \cdot 3^{\frac{2}{2}}}$

Step 1.3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.7.4.1

Cancel the common factor.

$\cos (θ) = \frac{21 \sqrt{3}}{16 \cdot 3^{\frac{2}{2}}}$

Step 1.3.7.4.2

Rewrite the expression.

$\cos (θ) = \frac{21 \sqrt{3}}{16 \cdot 3^{1}}$

Step 1.3.7.5

Evaluate the exponent.

$\cos (θ) = \frac{21 \sqrt{3}}{16 \cdot 3}$

Step 1.4

Cancel the common factor of $21$ and $3$ .

Tap for more steps...

Step 1.4.1

Factor $3$ out of $21 \sqrt{3}$ .

$\cos (θ) = \frac{3 (7 \sqrt{3})}{16 \cdot 3}$

Step 1.4.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.1

Factor $3$ out of $16 \cdot 3$ .

$\cos (θ) = \frac{3 (7 \sqrt{3})}{3 \cdot 16}$

Step 1.4.2.2

Cancel the common factor.

$\cos (θ) = \frac{3 (7 \sqrt{3})}{3 \cdot 16}$

Step 1.4.2.3

Rewrite the expression.

$\cos (θ) = \frac{7 \sqrt{3}}{16}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$θ \approx 0.71090412 + 2 π n, 5.57228118 + 2 π n$ , for any integer $n$

Step 3