Trigonometry Examples

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

Step 1

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

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

Adjacent : $- \frac{\sqrt{2}}{2}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

Simplify the expression.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

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

Step 2.3.4.2

Rewrite the expression.

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

Step 2.3.5

Evaluate the exponent.

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $2$ and $4$ .

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Step 2.5.1

Factor $2$ out of $2$ .

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

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $4$ .

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

Step 2.5.2.2

Cancel the common factor.

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

Step 2.5.2.3

Rewrite the expression.

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

Step 2.6

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

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

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

Step 2.7.4.2

Rewrite the expression.

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

Step 2.7.5

Evaluate the exponent.

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

Step 2.8

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

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

Step 2.9

Cancel the common factor of $2$ and $4$ .

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Step 2.9.1

Factor $2$ out of $2$ .

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

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Factor $2$ out of $4$ .

$\sqrt{\frac{1}{2} + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.9.2.2

Cancel the common factor.

$\sqrt{\frac{1}{2} + \frac{2 \cdot 1}{2 \cdot 2}}$

Step 2.9.2.3

Rewrite the expression.

$\sqrt{\frac{1}{2} + \frac{1}{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 1}{2}}$

Step 2.10.2

Add $1$ and $1$ .

$\sqrt{\frac{2}{2}}$

Step 2.10.3

Divide $2$ by $2$ .

$\sqrt{1}$

Step 2.10.4

Any root of $1$ is $1$ .

$1$

Step 3

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

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

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = 1 (\frac{2}{\sqrt{2}})$

Step 4.2

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

$\csc (θ) = \frac{2}{\sqrt{2}}$

Step 4.3

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

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

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

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

Step 4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

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

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Step 4.4.6.1

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

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

$\csc (θ) = \frac{2 \sqrt{2}}{2}$

Step 4.4.6.5

Evaluate the exponent.

$\csc (θ) = \frac{2 \sqrt{2}}{2}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$\csc (θ) = \frac{2 \sqrt{2}}{2}$

Step 4.5.2

Divide $\sqrt{2}$ by $1$ .

$\csc (θ) = \sqrt{2}$

Step 5

Approximate the result.

$\csc (θ) = \sqrt{2} \approx 1.41421356$