Problem 33
Question
Solve the equation. $$3 \sec ^{2} x-4=0$$
Step-by-Step Solution
Verified Answer
The equation \(3 \sec ^{2} x-4=0\) has no solution.
1Step 1: Isolate secant squared term
Isolate \(3 \sec ^{2} x\) by moving 4 to the right side of the equation: \(3 \sec ^{2} x = 4\).
2Step 2: Isolate secant term
Next, to isolate \( \sec ^{2} x \), divide both sides of the equation by 3: \( \sec ^{2} x = \frac{4}{3}\).
3Step 3: Take the square root
Taking square root of both sides gives \( \sec x = \pm \sqrt{\frac{4}{3}} \), since square root has both positive and negative values.
4Step 4: Find the value of x
Taking the arcsecant (\(\text{sec}^{-1}\)) of both sides will give the value of \(x\). It is understood that \( \sec = \frac{1}{\cos}\), thus \( \cos x = \frac{3}{2} \) for \(x\). However, this value is not possible since the absolute value of cosine function is less than or equal to 1. Therefore, there is no solution.
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