Trigonometry Examples

$(\frac{p}{2}, \frac{p}{3})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{p}{2}, \frac{p}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{p}{2}, 0)$ , and $(\frac{p}{2}, \frac{p}{3})$ .

Opposite : $\frac{p}{3}$

Adjacent : $\frac{p}{2}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

Tap for more steps...

Step 2.1

Apply the product rule to $\frac{p}{2}$ .

$\sqrt{\frac{p^{2}}{2^{2}} + {(\frac{p}{3})}^{2}}$

Step 2.2

Raise $2$ to the power of $2$ .

$\sqrt{\frac{p^{2}}{4} + {(\frac{p}{3})}^{2}}$

Step 2.3

Apply the product rule to $\frac{p}{3}$ .

$\sqrt{\frac{p^{2}}{4} + \frac{p^{2}}{3^{2}}}$

Step 2.4

Raise $3$ to the power of $2$ .

$\sqrt{\frac{p^{2}}{4} + \frac{p^{2}}{9}}$

Step 2.5

To write $\frac{p^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{p^{2}}{4} \cdot \frac{9}{9} + \frac{p^{2}}{9}}$

Step 2.6

To write $\frac{p^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{p^{2}}{4} \cdot \frac{9}{9} + \frac{p^{2}}{9} \cdot \frac{4}{4}}$

Step 2.7

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.7.1

Multiply $\frac{p^{2}}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{p^{2} \cdot 9}{4 \cdot 9} + \frac{p^{2}}{9} \cdot \frac{4}{4}}$

Step 2.7.2

Multiply $4$ by $9$ .

$\sqrt{\frac{p^{2} \cdot 9}{36} + \frac{p^{2}}{9} \cdot \frac{4}{4}}$

Step 2.7.3

Multiply $\frac{p^{2}}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{p^{2} \cdot 9}{36} + \frac{p^{2} \cdot 4}{9 \cdot 4}}$

Step 2.7.4

Multiply $9$ by $4$ .

$\sqrt{\frac{p^{2} \cdot 9}{36} + \frac{p^{2} \cdot 4}{36}}$

Step 2.8

Combine the numerators over the common denominator.

$\sqrt{\frac{p^{2} \cdot 9 + p^{2} \cdot 4}{36}}$

Step 2.9

Rewrite $\frac{p^{2} \cdot 9 + p^{2} \cdot 4}{36}$ in a factored form.

Tap for more steps...

Step 2.9.1

Move $9$ to the left of $p^{2}$ .

$\sqrt{\frac{9 \cdot p^{2} + p^{2} \cdot 4}{36}}$

Step 2.9.2

Move $4$ to the left of $p^{2}$ .

$\sqrt{\frac{9 p^{2} + 4 \cdot p^{2}}{36}}$

Step 2.9.3

Add $9 p^{2}$ and $4 p^{2}$ .

$\sqrt{\frac{13 p^{2}}{36}}$

Step 2.10

Rewrite $\frac{13 p^{2}}{36}$ as ${(\frac{p}{6})}^{2} \cdot 13$ .

Tap for more steps...

Step 2.10.1

Factor the perfect power $p^{2}$ out of $13 p^{2}$ .

$\sqrt{\frac{p^{2} \cdot 13}{36}}$

Step 2.10.2

Factor the perfect power $6^{2}$ out of $36$ .

$\sqrt{\frac{p^{2} \cdot 13}{6^{2} \cdot 1}}$

Step 2.10.3

Rearrange the fraction $\frac{p^{2} \cdot 13}{6^{2} \cdot 1}$ .

$\sqrt{{(\frac{p}{6})}^{2} \cdot 13}$

Step 2.11

Pull terms out from under the radical.

$\frac{p}{6} \sqrt{13}$

Step 2.12

Combine $\frac{p}{6}$ and $\sqrt{13}$ .

$\frac{p \sqrt{13}}{6}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{p}{2}}{\frac{p \sqrt{13}}{6}}$ .

$\frac{\frac{p}{2}}{\frac{p \sqrt{13}}{6}}$

Step 4

Simplify $\cos (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{p}{2} \cdot \frac{6}{p \sqrt{13}}$

Step 4.2

Cancel the common factor of $p$ .

Tap for more steps...

Step 4.2.1

Factor $p$ out of $p \sqrt{13}$ .

$\cos (θ) = \frac{p}{2} \cdot \frac{6}{p (\sqrt{13})}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{p}{2} \cdot \frac{6}{p \sqrt{13}}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = \frac{1}{2} \cdot \frac{6}{\sqrt{13}}$

Step 4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.1

Factor $2$ out of $6$ .

$\cos (θ) = \frac{1}{2} \cdot \frac{2 (3)}{\sqrt{13}}$

Step 4.3.2

Cancel the common factor.

$\cos (θ) = \frac{1}{2} \cdot \frac{2 \cdot 3}{\sqrt{13}}$

Step 4.3.3

Rewrite the expression.

$\cos (θ) = \frac{3}{\sqrt{13}}$

Step 4.4

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = \frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Step 4.5

Combine and simplify the denominator.

Tap for more steps...

Step 4.5.1

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.5.2

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.5.3

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.5.4

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

$\cos (θ) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\cos (θ) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Tap for more steps...

Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\cos (θ) = \frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\cos (θ) = \frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\cos (θ) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{3 \sqrt{13}}{13}$

Step 4.5.6.5

Evaluate the exponent.

$\cos (θ) = \frac{3 \sqrt{13}}{13}$

Step 5

Approximate the result.

$\cos (θ) = \frac{3 \sqrt{13}}{13} \approx 0.83205029$