Trigonometry Examples

$\sin (θ) = - \frac{\sqrt{2}}{2}$

Step 1

To find the value of $\csc (θ)$ , use the fact that $\frac{1}{\sin (θ)}$ then substitute in the known values.

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

Step 2

Simplify the result.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Move the negative in front of the fraction.

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

Step 2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

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

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

Step 2.5.5

Add $1$ and $1$ .

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

Step 2.5.6

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

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Step 2.5.6.1

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

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

Step 2.5.6.2

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

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

Step 2.5.6.3

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

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

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

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

Step 2.5.6.4.2

Rewrite the expression.

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

Step 2.5.6.5

Evaluate the exponent.

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

Step 2.6

Cancel the common factor of $2$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

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

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