Find the Exact Value sec(585 degrees )

$\sec (585 °)$

Step 1

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sec (225)$

Step 2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the third quadrant.

$- \sec (45)$

Step 3

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

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

Step 4

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

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

Step 5

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.

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

Step 6

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

$- \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$

Find the Exact Value sec(585 degrees )

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