Question

Solve each equation for exact solutions.arcsin 2x + arccos x = π/6

Accepted Answer

Recall the identity that relates arcsin and arccos: for any value $$t in $$the domain, $$\arcsin(t) + \arccos(t) = \frac{\pi}{2}. $$This will help simplify the given equation. Rewrite the given equation $$\arcsin(2x) + \arccos(x) = \frac{\pi}{6}$$ and consider how to use the identity. Since the identity involves the same argument, but here the arguments are $$2x$$ and $$x$$, think about expressing one inverse function in terms of the other or isolating one term. Isolate one of the inverse trigonometric functions, for example, write $$\arcsin(2x) = \frac{\pi}{6} - \arccos(x). $$Apply the sine function to both sides to eliminate the arcsin on the left: $$\sin(\arcsin(2x)) = \sin\left(\frac{\pi}{6} - \arccos(x)\right). $$This simplifies to $$2x = \sin\left(\frac{\pi}{6} - \arccos(x)\right). $$Use the sine difference formula $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ with $$a = \frac{\pi}{6}$$ and $$b = \arccos(x) to $$expand the right side. Then substitute $$\cos(\arccos(x)) = x$$ and $$\sin(\arccos(x)) = \sqrt{1 - x^2$} to $$get an equation in terms of $$x$$ only.

Solve each equation for exact solutions.
arcsin 2x + arccos x = π/6

주요 개념

Inverse Trigonometric Functions

Domain and Range Restrictions

Trigonometric Identities and Equation Manipulation