Trigonometry Examples

$(x, y)$

Step 1

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

Opposite : $y$

Adjacent : $x$

Step 2

Remove parentheses.

$\sqrt{x^{2} + y^{2}}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{y}{\sqrt{x^{2} + y^{2}}}$ .

$\frac{y}{\sqrt{x^{2} + y^{2}}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{y}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$\sin (θ) = \frac{y}{\sqrt{x^{2} + y^{2}}} \cdot \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{y}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2}}}$

Step 4.2.2

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2}}}$

Step 4.2.3

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2}}}$

Step 4.2.4

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

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{\sqrt{x^{2} + y^{2}}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{x^{2} + y^{2}}}^{2}$ as $x^{2} + y^{2}$ .

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Step 4.2.6.1

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

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{({(x^{2} + y^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{{(x^{2} + y^{2})}^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

Step 4.2.6.5

Simplify.

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$

Step 5

Approximate the result.

$\sin (θ) = \frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}} \approx \frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$