Trigonometry Examples

$(0, \frac{9}{41})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(0, \frac{9}{41})$ , draw the triangle between the three points $(0, 0)$ , $(0, 0)$ , and $(0, \frac{9}{41})$ .

Opposite : $\frac{9}{41}$

Adjacent : $0$

Step 2

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

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Step 2.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + {(\frac{9}{41})}^{2}}$

Step 2.2

Apply the product rule to $\frac{9}{41}$ .

$\sqrt{0 + \frac{9^{2}}{41^{2}}}$

Step 2.3

Raise $9$ to the power of $2$ .

$\sqrt{0 + \frac{81}{41^{2}}}$

Step 2.4

Raise $41$ to the power of $2$ .

$\sqrt{0 + \frac{81}{1681}}$

Step 2.5

Add $0$ and $\frac{81}{1681}$ .

$\sqrt{\frac{81}{1681}}$

Step 2.6

Rewrite $\sqrt{\frac{81}{1681}}$ as $\frac{\sqrt{81}}{\sqrt{1681}}$ .

$\frac{\sqrt{81}}{\sqrt{1681}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Rewrite $81$ as $9^{2}$ .

$\frac{\sqrt{9^{2}}}{\sqrt{1681}}$

Step 2.7.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{9}{\sqrt{1681}}$

Step 2.8

Simplify the denominator.

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Step 2.8.1

Rewrite $1681$ as $41^{2}$ .

$\frac{9}{\sqrt{41^{2}}}$

Step 2.8.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{9}{41}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{9}{41}}{\frac{9}{41}}$ .

$\frac{\frac{9}{41}}{\frac{9}{41}}$

Step 4

Cancel the common factor of $\frac{9}{41}$ .

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Step 4.1

Cancel the common factor.

$\sin (θ) = \frac{\frac{9}{41}}{\frac{9}{41}}$

Step 4.2

Rewrite the expression.

$\sin (θ) = 1$