How do you find two solutions (in degree and radians) for $sec x = \sqrt{2}$ $secx = sqrt2$ ?

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How do you find two solutions (in degree and radians) for $sec x = \sqrt{2}$ $secx = sqrt2$ ?

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Answer 1 · 2015-10-13T14:31:03

The two solutions in degrees (from 0 to 360) would be 45 and 315.
In radians (from 0 to $2 π$ $2pi$ ) the answer would be $\frac{π}{4}$ $pi/4$ and $\frac{7 π}{4}$ $(7pi)/4$

Explanation:

$sec (x) = \frac{1}{cos (x)}$ $sec(x) = 1/cos(x)$

Plug in our givens:

$\sqrt{2} = \frac{1}{cos (x)}$ $sqrt(2) = 1/cos(x)$
$cos (x) = \frac{1}{\sqrt{2}}$ $cos(x) = 1/sqrt(2)$
$cos (x) = \frac{\sqrt{2}}{2}$ $cos(x) = sqrt(2)/2$

Using the unit circle, I know that $cos (45)$ $cos(45)$ or $cos (\frac{π}{4})$ $cos(pi/4)$ are equal to $\frac{\sqrt{2}}{2}$ $sqrt(2)/2$ , and therefore I can determine that 45 or $\frac{π}{4}$ $pi/4$ are one of the angles.

We also know that the value at $- \frac{π}{4}$ $-pi/4$ or $- 45$ $-45$ are equal to $\frac{\sqrt{2}}{2}$ $sqrt(2)/2$ , and so we can have this as our second angle ( $- \frac{π}{4}$ $-pi/4$ can be written as a negative, or as $\frac{7 π}{4}$ $(7pi)/4$ and same goes for -45, which would instead be 315.)

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How do you find two solutions (in degree and radians) for $sec x = \sqrt{2}$ $secx = sqrt2$ ?

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How do you find two solutions (in degree and radians) for $sec x = \sqrt{2}$ $secx = sqrt2$ ?