Question

Solve each equation for x, where x is restricted to the given interval.y = √2 + 3 sec 2x, for x in [0, π/4) ⋃ (π/4, π/2]

Accepted Answer

Rewrite the given equation: $$y = \sqrt{2} + 3 \sec 2x. $$Our goal is to solve for $$x in $$the interval $$[0, \frac{\pi}{4}) \cup (\frac{\pi}{4}, \frac{\pi}{2}]. $$Isolate the $$\sec 2x$$ term by subtracting $$\sqrt{2}$$ from both sides: $$y - \sqrt{2} = 3 \sec 2x. $$Then divide both sides by 3 to get $$\sec 2x = \frac{y - \sqrt{2}}{3}. $$Recall that $$\sec 	heta = \frac{1}{\cos 	heta}. So$$, rewrite the equation as $$\frac{1}{\cos 2x} = \frac{y - \sqrt{2}}{3}$$, which implies $$\cos 2x = \frac{3}{y - \sqrt{2}}. $$Solve for $$2x by $$taking the inverse cosine (arccos) of both sides: $$2x = \arccos \left( \frac{3}{y - \sqrt{2}} ight). $$Remember that cosine is positive in the first and fourth quadrants, so consider all possible solutions for $$2x$$ within the domain. Finally, divide by 2 to solve for $$x$$: $$x = \frac{1}{2} \arccos \left( \frac{3}{y - \sqrt{2}} ight). $$Check which solutions fall within the given interval $$[0, \frac{\pi}{4}) \cup (\frac{\pi}{4}, \frac{\pi}{2}]$$ and exclude any values where $$\sec 2x is $$undefined (such as at $$x = \frac{\pi}{4}).$$

Solve each equation for x, where x is restricted to the given interval.
y = √2 + 3 sec 2x, for x in [0, π/4) ⋃ (π/4, π/2]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Secant Function and Its Properties

Solving Trigonometric Equations

Domain Restrictions and Interval Notation