QuestionFind all $\theta$ in $[0^{\circ}, 360^{\circ})$ such that $\sec \theta = -\sqrt{2}$ . What are the values of $\theta$ ?

Studdy Solution

STEP 1

Assumptions1. The function value of $\sec \theta$ is given as $-\sqrt{}$ . . The range of $\theta$ is between $0^{\circ}$ and $360^{\circ}$ , inclusive of $0^{\circ}$ and exclusive of $360^{\circ}$ .
3. The secant function $\sec \theta$ is defined as the reciprocal of the cosine function, i.e., $\sec \theta = \frac{1}{\cos \theta}$ .
4. The cosine function $\cos \theta$ is negative in the second and third quadrants of the unit circle.

STEP 2

First, we need to find the value of $\cos \theta$ from the given value of $\sec \theta$ . We can do this by taking the reciprocal of $\sec \theta$ .
$\cos \theta = \frac{1}{\sec \theta}$

STEP 3

Now, plug in the given value for $\sec \theta$ to calculate $\cos \theta$ .
$\cos \theta = \frac{1}{-\sqrt{2}}$

STEP 4

Calculate the value of $\cos \theta$ .
$\cos \theta = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$

STEP 5

Now that we have the value of $\cos \theta$ , we need to find the angles $\theta$ in the interval $[0^{\circ},360^{\circ})$ for which $\cos \theta = -\frac{\sqrt{2}}{2}$ .
The angles for which $\cos \theta = \pm \frac{\sqrt{2}}{2}$ are well-known and correspond to $\theta =45^{\circ},135^{\circ},225^{\circ},$ and $315^{\circ}$ .

STEP 6

However, since $\cos \theta$ is negative, we only consider the angles in the second and third quadrants. Therefore, the solutions to $\cos \theta = -\frac{\sqrt{2}}{2}$ are $\theta =135^{\circ},225^{\circ}$ So, the values of $\theta$ that satisfy $\sec \theta = -\sqrt{2}$ are $135^{\circ}$ and $225^{\circ}$ .

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