Trigonometry Examples

$(6, - 8)$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(6, - 8)$ , draw the triangle between the three points $(0, 0)$ , $(6, 0)$ , and $(6, - 8)$ .

Opposite : $- 8$

Adjacent : $6$

Step 2

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

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Step 2.1

Raise $6$ to the power of $2$ .

$\sqrt{36 + {(- 8)}^{2}}$

Step 2.2

Raise $- 8$ to the power of $2$ .

$\sqrt{36 + 64}$

Step 2.3

Add $36$ and $64$ .

$\sqrt{100}$

Step 2.4

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}}$

Step 2.5

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

$10$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{10}{- 8}$ .

$\frac{10}{- 8}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Cancel the common factor of $10$ and $- 8$ .

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Step 4.1.1

Factor $2$ out of $10$ .

$\csc (θ) = \frac{2 (5)}{- 8}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $- 8$ .

$\csc (θ) = \frac{2 \cdot 5}{2 \cdot - 4}$

Step 4.1.2.2

Cancel the common factor.

$\csc (θ) = \frac{2 \cdot 5}{2 \cdot - 4}$

Step 4.1.2.3

Rewrite the expression.

$\csc (θ) = \frac{5}{- 4}$

Step 4.2

Move the negative in front of the fraction.

$\csc (θ) = - \frac{5}{4}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{5}{4} \approx - 1.25$