Trigonometry Examples

$(\sqrt{5}, 2)$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\sqrt{5}, 2)$ , draw the triangle between the three points $(0, 0)$ , $(\sqrt{5}, 0)$ , and $(\sqrt{5}, 2)$ .

Opposite : $2$

Adjacent : $\sqrt{5}$

Step 2

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

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Step 2.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{{(5^{\frac{1}{2}})}^{2} + {(2)}^{2}}$

Step 2.1.2

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

$\sqrt{5^{\frac{1}{2} \cdot 2} + {(2)}^{2}}$

Step 2.1.3

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

$\sqrt{5^{\frac{2}{2}} + {(2)}^{2}}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\sqrt{5^{\frac{2}{2}} + {(2)}^{2}}$

Step 2.1.4.2

Rewrite the expression.

$\sqrt{5^{1} + {(2)}^{2}}$

Step 2.1.5

Evaluate the exponent.

$\sqrt{5 + {(2)}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

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

$\sqrt{5 + 4}$

Step 2.2.2

Add $5$ and $4$ .

$\sqrt{9}$

Step 2.2.3

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 2.3

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

$3$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{3}{2}$ .

$\frac{3}{2}$

Step 4

Approximate the result.

$\csc (θ) = \frac{3}{2} \approx 1.5$