Find the Value Using the Unit Circle csc(315)

$\csc (315)$

Step 1

Find the value using the definition of cosecant.

$\csc (315) = \frac{hypotenuse}{opposite}$

Step 2

Substitute the values into the definition.

$\csc (315) = \frac{1}{- \frac{\sqrt{2}}{2}}$

Step 3

Simplify the result.

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

$- \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$

Step 5

Find the Value Using the Unit Circle csc(315)

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