# Find the two possible values of theta

• Jun 8th 2010, 04:27 PM
RogueDemon
Find the two possible values of theta
$\displaystyle sec{\Theta} = -1.942$
$\displaystyle \frac{1}{cos\Theta} = -1.942$
$\displaystyle \frac{1}{-1.942} = cos\Theta$
$\displaystyle cos^{-1}\frac{1}{-1.942} = \Theta$
$\displaystyle 121^{\circ} = \Theta$

Now that I've got the first value, how do I find the second? All I know is that the CAST rule may be involved, but I don't know how to apply it in this situation. Help on this would be greatly appreciated.
• Jun 8th 2010, 04:36 PM
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Quote:

Originally Posted by RogueDemon
$\displaystyle sec{\Theta} = -1.942$
$\displaystyle \frac{1}{cos\Theta} = -1.942$
$\displaystyle \frac{1}{-1.942} = cos\Theta$
$\displaystyle cos^{-1}\frac{1}{-1.942} = \Theta$
$\displaystyle 121^{\circ} = \Theta$

Now that I've got the first value, how do I find the second? All I know is that the CAST rule may be involved, but I don't know how to apply it in this situation. Help on this would be greatly appreciated.

Assuming you don't need any more precision than what you indicated, the other result is given by $\displaystyle (360 - 121)^{\circ} = 239^{\circ}$

Note that there are actually an infinite number of possible values given by $\displaystyle \theta = \theta_0 + k\cdot 360^{\circ}, k \in \mathbb{Z}$ where $\displaystyle \theta_0=121^{\circ}$ or $\displaystyle \theta_0=239^{\circ}$.
• Jun 8th 2010, 04:43 PM
RogueDemon
Thanks very much for the response. Just wondering though, why is it 360 - 121 as opposed to something like 180 - 121? Is it because this question involves cos, where cos covers the 360 degree region of the circle and sin covers the 180 degree region?
• Jun 8th 2010, 04:54 PM
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Quote:

Originally Posted by RogueDemon
Thanks very much for the response. Just wondering though, why is it 360 - 121 as opposed to something like 180 - 121? Is it because this question involves cos, where cos covers the 360 degree region of the circle and sin covers the 180 degree region?

Actually it's because cos(x) is an even function, meaning cos(x) = cos(-x). It just so happens that cos(-x) = cos(360 - x). I figured you were probably looking for an answer in the range [0, 360], so I gave the latter answer.

The corresponding identity for sin(x) is sin(x) = -sin(-x) because sin(x) is odd.

It is also true that sin(x) = sin(180-x) but this more closely corresponds with cos(x) = -cos(180-x). See here.