Trigonometry Examples

$(3 \frac{1}{2}, - 2 \sqrt{15})$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(3 \frac{1}{2}, - 2 \sqrt{15})$ , draw the triangle between the three points $(0, 0)$ , $(3 \frac{1}{2}, 0)$ , and $(3 \frac{1}{2}, - 2 \sqrt{15})$ .

Opposite : $- 2 \sqrt{15}$

Adjacent : $3 \frac{1}{2}$

Step 2

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

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Step 2.1

Convert $3 \frac{1}{2}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$\sqrt{{(3 + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2

Add $3$ and $\frac{1}{2}$ .

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Step 2.1.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{{(3 \cdot \frac{2}{2} + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.2

Combine $3$ and $\frac{2}{2}$ .

$\sqrt{{(\frac{3 \cdot 2}{2} + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\sqrt{{(\frac{3 \cdot 2 + 1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $3$ by $2$ .

$\sqrt{{(\frac{6 + 1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.4.2

Add $6$ and $1$ .

$\sqrt{{(\frac{7}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $\frac{7}{2}$ .

$\sqrt{\frac{7^{2}}{2^{2}} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.2

Raise $7$ to the power of $2$ .

$\sqrt{\frac{49}{2^{2}} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.3

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

$\sqrt{\frac{49}{4} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.4

Apply the product rule to $- 2 \sqrt{15}$ .

$\sqrt{\frac{49}{4} + {(- 2)}^{2} {\sqrt{15}}^{2}}$

Step 2.2.5

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

$\sqrt{\frac{49}{4} + 4 {\sqrt{15}}^{2}}$

Step 2.3

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sqrt{\frac{49}{4} + 4 {(15^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\sqrt{\frac{49}{4} + 4 \cdot 15^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\sqrt{\frac{49}{4} + 4 \cdot 15^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{\frac{49}{4} + 4 \cdot 15^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{\frac{49}{4} + 4 \cdot 15^{1}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{\frac{49}{4} + 4 \cdot 15}$

Step 2.4

Multiply $4$ by $15$ .

$\sqrt{\frac{49}{4} + 60}$

Step 2.5

To write $60$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{49}{4} + 60 \cdot \frac{4}{4}}$

Step 2.6

Combine $60$ and $\frac{4}{4}$ .

$\sqrt{\frac{49}{4} + \frac{60 \cdot 4}{4}}$

Step 2.7

Combine the numerators over the common denominator.

$\sqrt{\frac{49 + 60 \cdot 4}{4}}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $60$ by $4$ .

$\sqrt{\frac{49 + 240}{4}}$

Step 2.8.2

Add $49$ and $240$ .

$\sqrt{\frac{289}{4}}$

Step 2.9

Rewrite $\sqrt{\frac{289}{4}}$ as $\frac{\sqrt{289}}{\sqrt{4}}$ .

$\frac{\sqrt{289}}{\sqrt{4}}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Rewrite $289$ as $17^{2}$ .

$\frac{\sqrt{17^{2}}}{\sqrt{4}}$

Step 2.10.2

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

$\frac{17}{\sqrt{4}}$

Step 2.11

Simplify the denominator.

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Step 2.11.1

Rewrite $4$ as $2^{2}$ .

$\frac{17}{\sqrt{2^{2}}}$

Step 2.11.2

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

$\frac{17}{2}$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{\frac{17}{2}}{- 2 \sqrt{15}}$ .

$\frac{\frac{17}{2}}{- 2 \sqrt{15}}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = \frac{17}{2} \cdot \frac{1}{- 2 \sqrt{15}}$

Step 4.2

Move the negative in front of the fraction.

$\csc (θ) = \frac{17}{2} \cdot (- \frac{1}{2 \sqrt{15}})$

Step 4.3

Multiply $\frac{1}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\csc (θ) = \frac{17}{2} \cdot (- (\frac{1}{2 \sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}))$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \sqrt{15} \sqrt{15}})$

Step 4.4.2

Move $\sqrt{15}$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 4.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 4.4.4

Raise $\sqrt{15}$ to the power of $1$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 4.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 {\sqrt{15}}^{1 + 1}})$

Step 4.4.6

Add $1$ and $1$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 {\sqrt{15}}^{2}})$

Step 4.4.7

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 {(15^{\frac{1}{2}})}^{2}})$

Step 4.4.7.2

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

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{1}{2} \cdot 2}})$

Step 4.4.7.3

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

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 4.4.7.4

Cancel the common factor of $2$ .

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Step 4.4.7.4.1

Cancel the common factor.

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 4.4.7.4.2

Rewrite the expression.

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15})$

Step 4.4.7.5

Evaluate the exponent.

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{2 \cdot 15})$

Step 4.5

Multiply $2$ by $15$ .

$\csc (θ) = \frac{17}{2} \cdot (- \frac{\sqrt{15}}{30})$

Step 4.6

Multiply $\frac{17}{2} (- \frac{\sqrt{15}}{30})$ .

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Step 4.6.1

Multiply $\frac{17}{2}$ by $\frac{\sqrt{15}}{30}$ .

$\csc (θ) = - \frac{17 \sqrt{15}}{2 \cdot 30}$

Step 4.6.2

Multiply $2$ by $30$ .

$\csc (θ) = - \frac{17 \sqrt{15}}{60}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{17 \sqrt{15}}{60} \approx - 1.09734528$