Trigonometry Examples

$(\frac{2}{19}, y)$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{2}{19}, y)$ , draw the triangle between the three points $(0, 0)$ , $(\frac{2}{19}, 0)$ , and $(\frac{2}{19}, y)$ .

Opposite : $y$

Adjacent : $\frac{2}{19}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{2}{19}$ .

$\sqrt{\frac{2^{2}}{19^{2}} + {(y)}^{2}}$

Step 2.2

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

$\sqrt{\frac{4}{19^{2}} + {(y)}^{2}}$

Step 2.3

Raise $19$ to the power of $2$ .

$\sqrt{\frac{4}{361} + {(y)}^{2}}$

Step 2.4

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{361}{361}$ .

$\sqrt{\frac{4}{361} + y^{2} \cdot \frac{361}{361}}$

Step 2.5

Simplify terms.

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Step 2.5.1

Combine $y^{2}$ and $\frac{361}{361}$ .

$\sqrt{\frac{4}{361} + \frac{y^{2} \cdot 361}{361}}$

Step 2.5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + y^{2} \cdot 361}{361}}$

Step 2.6

Move $361$ to the left of $y^{2}$ .

$\sqrt{\frac{4 + 361 y^{2}}{361}}$

Step 2.7

Rewrite $\frac{4 + 361 y^{2}}{361}$ as ${(\frac{1}{19})}^{2} (4 + 361 y^{2})$ .

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Step 2.7.1

Factor the perfect power $1^{2}$ out of $4 + 361 y^{2}$ .

$\sqrt{\frac{1^{2} (4 + 361 y^{2})}{361}}$

Step 2.7.2

Factor the perfect power $19^{2}$ out of $361$ .

$\sqrt{\frac{1^{2} (4 + 361 y^{2})}{19^{2} \cdot 1}}$

Step 2.7.3

Rearrange the fraction $\frac{1^{2} (4 + 361 y^{2})}{19^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{19})}^{2} (4 + 361 y^{2})}$

Step 2.8

Pull terms out from under the radical.

$\frac{1}{19} \sqrt{4 + 361 y^{2}}$

Step 2.9

Combine $\frac{1}{19}$ and $\sqrt{4 + 361 y^{2}}$ .

$\frac{\sqrt{4 + 361 y^{2}}}{19}$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{\frac{\sqrt{4 + 361 y^{2}}}{19}}{y}$ .

$\frac{\frac{\sqrt{4 + 361 y^{2}}}{19}}{y}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = \frac{\sqrt{4 + 361 y^{2}}}{19} \cdot \frac{1}{y}$

Step 4.2

Combine.

$\csc (θ) = \frac{\sqrt{4 + 361 y^{2}} \cdot 1}{19 y}$

Step 4.3

Multiply $\sqrt{4 + 361 y^{2}}$ by $1$ .

$\csc (θ) = \frac{\sqrt{4 + 361 y^{2}}}{19 y}$

Step 5

Approximate the result.

$\csc (θ) = \frac{\sqrt{4 + 361 y^{2}}}{19 y} \approx \frac{\sqrt{4 + 361 y^{2}}}{19 y}$