Answer:
[tex]\theta = \dfrac{\pi}{4}[/tex]
Step-by-step explanation:
We are solving for θ in the equation:
[tex](\sec(\theta))^2 - 6 = -4[/tex]
First, we can add 6 to both sides.
[tex](\sec(\theta))^2 = -4+6[/tex]
[tex](\sec(\theta))^2 = 2[/tex]
Then, we can take the square root of both sides:
[tex]\sec(\theta) = \sqrt2[/tex]
Next, we can replace secant with its definition:
[tex]\dfrac{1}{\cos(\theta)} = \sqrt2[/tex]
To isolate cosine, we can now take the reciprocal of both sides:
[tex]\cos(\theta)=\dfrac{1}{\sqrt2}[/tex]
Finally, we can identify the angle for which cosine (adjacent over hypotenuse) is 1 / √2. This angle is π/4 or 45°.
[tex]\boxed{\theta = \dfrac{\pi}{4}}[/tex]
Further Note
We can also rationalize the value of cosine to make it look the same as the unit circle value:
[tex]\cos\!\left(\dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt2} \cdot \dfrac{\sqrt2}{\sqrt2} = \dfrac{\sqrt{2}}2[/tex]
