Trigonometry Examples

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

Step 1

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

Opposite : $- 1$

Adjacent : $- \sqrt{3}$

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 $- \sqrt{3}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Evaluate the exponent.

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

Step 2.3

Simplify the expression.

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Step 2.3.1

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

$\sqrt{3 + 1}$

Step 2.3.2

Add $3$ and $1$ .

$\sqrt{4}$

Step 2.3.3

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

$\sqrt{2^{2}}$

Step 2.4

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

$2$

Step 3

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

$\frac{2}{- 1}$

Step 4

Divide $2$ by $- 1$ .

$\csc (θ) = - 2$