Certainly! Let's solve the equation step-by-step:
Given the equation:
[tex]\[ 7 \sec^2 \theta + \tan^2 \theta = 15 \][/tex]
First, recall that the secant function and the tangent function have a trigonometric identity relationship. Specifically, for any angle [tex]\(\theta\)[/tex]:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]
We will use this identity to simplify our given equation. Substitute [tex]\(\sec^2 \theta\)[/tex] with [tex]\(1 + \tan^2 \theta\)[/tex]:
[tex]\[ 7 (1 + \tan^2 \theta) + \tan^2 \theta = 15 \][/tex]
Expand and combine like terms:
[tex]\[ 7 + 7 \tan^2 \theta + \tan^2 \theta = 15 \][/tex]
[tex]\[ 7 + 8 \tan^2 \theta = 15 \][/tex]
Isolate the term involving [tex]\(\tan^2 \theta\)[/tex]:
[tex]\[ 8 \tan^2 \theta = 15 - 7 \][/tex]
[tex]\[ 8 \tan^2 \theta = 8 \][/tex]
Now divide both sides by 8:
[tex]\[ \tan^2 \theta = 1 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ \tan \theta = \pm 1 \][/tex]
We now need to determine the angles [tex]\(\theta\)[/tex] where the tangent is [tex]\(\pm 1\)[/tex]. Recall the values of [tex]\(\theta\)[/tex] in one full circle (from [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex] radians) where this occurs:
1. [tex]\(\tan \theta = 1\)[/tex] at [tex]\(\theta = \frac{\pi}{4} \)[/tex] and [tex]\(\theta = \frac{5\pi}{4}\)[/tex].
2. [tex]\(\tan \theta = -1\)[/tex] at [tex]\(\theta = \frac{3\pi}{4}\)[/tex] and [tex]\(\theta = \frac{7\pi}{4}\)[/tex].
However, [tex]\(\tan \theta\)[/tex] repeats every [tex]\(\pi\)[/tex] radians. Therefore, we can simply use the general solutions for these values within the range of [tex]\(0\)[/tex] to [tex]\(2\pi\)[/tex]. Thus, the angles are:
[tex]\[ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4} \][/tex]
To also consider angles in other quadrants, let's include the negative rotations:
[tex]\[ \theta = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
Therefore, the complete set of solutions is:
[tex]\[ \theta = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \][/tex]
These are the angles [tex]\(\theta\)[/tex] where the given equation [tex]\(7 \sec^2 \theta + \tan^2 \theta = 15\)[/tex] holds true.
