Trigonometry Examples

$(2 \sqrt{5}, 4)$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(2 \sqrt{5}, 4)$ , draw the triangle between the three points $(0, 0)$ , $(2 \sqrt{5}, 0)$ , and $(2 \sqrt{5}, 4)$ .

Opposite : $4$

Adjacent : $2 \sqrt{5}$

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{2^{2} {\sqrt{5}}^{2} + {(4)}^{2}}$

Step 2.1.2

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

$\sqrt{4 {\sqrt{5}}^{2} + {(4)}^{2}}$

Step 2.2

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

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Step 2.2.1

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

$\sqrt{4 {(5^{\frac{1}{2}})}^{2} + {(4)}^{2}}$

Step 2.2.2

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

$\sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} + {(4)}^{2}}$

Step 2.2.3

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

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {(4)}^{2}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {(4)}^{2}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 5^{1} + {(4)}^{2}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{4 \cdot 5 + {(4)}^{2}}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $4$ by $5$ .

$\sqrt{20 + {(4)}^{2}}$

Step 2.3.2

Raise $4$ to the power of $2$ .

$\sqrt{20 + 16}$

Step 2.3.3

Add $20$ and $16$ .

$\sqrt{36}$

Step 2.3.4

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

Step 2.3.5

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

$6$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{6}{4}$ .

$\frac{6}{4}$

Step 4

Cancel the common factor of $6$ and $4$ .

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Step 4.1

Factor $2$ out of $6$ .

$\csc (θ) = \frac{2 (3)}{4}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $4$ .

$\csc (θ) = \frac{2 \cdot 3}{2 \cdot 2}$

Step 4.2.2

Cancel the common factor.

$\csc (θ) = \frac{2 \cdot 3}{2 \cdot 2}$

Step 4.2.3

Rewrite the expression.

$\csc (θ) = \frac{3}{2}$

Step 5

Approximate the result.

$\csc (θ) = \frac{3}{2} \approx 1.5$