Trigonometry Examples

$(2, - p)$

Step 1

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

Opposite : $- p$

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 $2$ .

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

Step 2.2

Apply the product rule to $- p$ .

$\sqrt{4 + {(- 1)}^{2} p^{2}}$

Step 2.3

Raise $- 1$ to the power of $2$ .

$\sqrt{4 + 1 p^{2}}$

Step 2.4

Multiply $p^{2}$ by $1$ .

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

Step 3

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

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

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{p}{\sqrt{4 + p^{2}}}$

Step 4.2

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

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

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{\sqrt{4 + p^{2}} \sqrt{4 + p^{2}}}$

Step 4.3.2

Raise $\sqrt{4 + p^{2}}$ to the power of $1$ .

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{\sqrt{4 + p^{2}} \sqrt{4 + p^{2}}}$

Step 4.3.3

Raise $\sqrt{4 + p^{2}}$ to the power of $1$ .

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{\sqrt{4 + p^{2}} \sqrt{4 + p^{2}}}$

Step 4.3.4

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

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{\sqrt{4 + p^{2}}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{\sqrt{4 + p^{2}}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{4 + p^{2}}}^{2}$ as $4 + p^{2}$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 + p^{2}}$ as ${(4 + p^{2})}^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{({(4 + p^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{(4 + p^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{(4 + p^{2})}^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{{(4 + p^{2})}^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{4 + p^{2}}$

Step 4.3.6.5

Simplify.

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{4 + p^{2}}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{p \sqrt{4 + p^{2}}}{4 + p^{2}} \approx - \frac{p \sqrt{4 + p^{2}}}{4 + p^{2}}$