Trigonometry Examples

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

Step 1

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

Opposite : $- \frac{\sqrt{3}}{2}$

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

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 $\frac{1}{2}$ .

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

Step 2.2

One to any power is one.

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

Step 2.3

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

$\sqrt{\frac{1}{4} + {(- \frac{\sqrt{3}}{2})}^{2}}$

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{1}{4} + 1 \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.5.2

Multiply $\frac{{\sqrt{3}}^{2}}{2^{2}}$ by $1$ .

$\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.6.3

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

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{4} + \frac{3^{1}}{2^{2}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1}{4} + \frac{3}{2^{2}}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

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

$\sqrt{\frac{1}{4} + \frac{3}{4}}$

Step 2.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 3}{4}}$

Step 2.7.3

Add $1$ and $3$ .

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

Step 2.7.4

Divide $4$ by $4$ .

$\sqrt{1}$

Step 2.7.5

Any root of $1$ is $1$ .

$1$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{- \frac{\sqrt{3}}{2}}$ .

$\frac{1}{- \frac{\sqrt{3}}{2}}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\csc (θ) = \frac{- 1 \cdot - 1}{- \frac{\sqrt{3}}{2}}$

Step 4.1.2

Move the negative in front of the fraction.

$\csc (θ) = - \frac{1}{\frac{\sqrt{3}}{2}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = - (1 (\frac{2}{\sqrt{3}}))$

Step 4.3

Multiply $\frac{2}{\sqrt{3}}$ by $1$ .

$\csc (θ) = - \frac{2}{\sqrt{3}}$

Step 4.4

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 4.5.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.5.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 4.5.4

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

$\csc (θ) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\csc (θ) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\csc (θ) = - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\csc (θ) = - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\csc (θ) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$\csc (θ) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\csc (θ) = - \frac{2 \sqrt{3}}{3}$

Step 4.5.6.5

Evaluate the exponent.

$\csc (θ) = - \frac{2 \sqrt{3}}{3}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{2 \sqrt{3}}{3} \approx - 1.15470053$