Trigonometry Examples

$(- 2, \frac{3}{4} p)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, \frac{3}{4} p)$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, \frac{3}{4} p)$ .

Opposite : $\frac{3}{4} p$

Adjacent : $- 2$

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine $\frac{3}{4}$ and $p$ .

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

Step 2.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1

Apply the product rule to $\frac{3 p}{4}$ .

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

Step 2.3.2

Apply the product rule to $3 p$ .

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

Step 2.4

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

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

Step 2.5

Raise $4$ to the power of $2$ .

$\sqrt{4 + \frac{9 p^{2}}{16}}$

Step 2.6

To write $4$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{4 \cdot \frac{16}{16} + \frac{9 p^{2}}{16}}$

Step 2.7

Combine $4$ and $\frac{16}{16}$ .

$\sqrt{\frac{4 \cdot 16}{16} + \frac{9 p^{2}}{16}}$

Step 2.8

Combine the numerators over the common denominator.

$\sqrt{\frac{4 \cdot 16 + 9 p^{2}}{16}}$

Step 2.9

Multiply $4$ by $16$ .

$\sqrt{\frac{64 + 9 p^{2}}{16}}$

Step 2.10

Rewrite $\frac{64 + 9 p^{2}}{16}$ as ${(\frac{1}{4})}^{2} (64 + 9 p^{2})$ .

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Step 2.10.1

Factor the perfect power $1^{2}$ out of $64 + 9 p^{2}$ .

$\sqrt{\frac{1^{2} (64 + 9 p^{2})}{16}}$

Step 2.10.2

Factor the perfect power $4^{2}$ out of $16$ .

$\sqrt{\frac{1^{2} (64 + 9 p^{2})}{4^{2} \cdot 1}}$

Step 2.10.3

Rearrange the fraction $\frac{1^{2} (64 + 9 p^{2})}{4^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{4})}^{2} (64 + 9 p^{2})}$

Step 2.11

Pull terms out from under the radical.

$\frac{1}{4} \sqrt{64 + 9 p^{2}}$

Step 2.12

Combine $\frac{1}{4}$ and $\sqrt{64 + 9 p^{2}}$ .

$\frac{\sqrt{64 + 9 p^{2}}}{4}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{3}{4} p}{\frac{\sqrt{64 + 9 p^{2}}}{4}}$ .

$\frac{\frac{3}{4} p}{\frac{\sqrt{64 + 9 p^{2}}}{4}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{3}{4} \cdot (p (\frac{4}{\sqrt{64 + 9 p^{2}}}))$

Step 4.2

Combine $\frac{3}{4}$ and $p$ .

$\sin (θ) = \frac{3 p}{4} \cdot \frac{4}{\sqrt{64 + 9 p^{2}}}$

Step 4.3

Combine.

$\sin (θ) = \frac{3 p \cdot 4}{4 \sqrt{64 + 9 p^{2}}}$

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Cancel the common factor.

$\sin (θ) = \frac{3 p \cdot 4}{4 \sqrt{64 + 9 p^{2}}}$

Step 4.4.2

Rewrite the expression.

$\sin (θ) = \frac{3 p}{\sqrt{64 + 9 p^{2}}}$

Step 4.5

Multiply $\frac{3 p}{\sqrt{64 + 9 p^{2}}}$ by $\frac{\sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}}}$ .

$\sin (θ) = \frac{3 p}{\sqrt{64 + 9 p^{2}}} \cdot \frac{\sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}}}$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{3 p}{\sqrt{64 + 9 p^{2}}}$ by $\frac{\sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}}}$ .

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}} \sqrt{64 + 9 p^{2}}}$

Step 4.6.2

Raise $\sqrt{64 + 9 p^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}} \sqrt{64 + 9 p^{2}}}$

Step 4.6.3

Raise $\sqrt{64 + 9 p^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{\sqrt{64 + 9 p^{2}} \sqrt{64 + 9 p^{2}}}$

Step 4.6.4

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

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{\sqrt{64 + 9 p^{2}}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{\sqrt{64 + 9 p^{2}}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{64 + 9 p^{2}}}^{2}$ as $64 + 9 p^{2}$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 + 9 p^{2}}$ as ${(64 + 9 p^{2})}^{\frac{1}{2}}$ .

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{({(64 + 9 p^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{(64 + 9 p^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{(64 + 9 p^{2})}^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{{(64 + 9 p^{2})}^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{64 + 9 p^{2}}$

Step 4.6.6.5

Simplify.

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{64 + 9 p^{2}}$

Step 5

Approximate the result.

$\sin (θ) = \frac{3 p \sqrt{64 + 9 p^{2}}}{64 + 9 p^{2}} \approx \frac{3 p \sqrt{64 + 9 p^{2}}}{64 + 9 p^{2}}$