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

Verified step by step guidance

1. 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.
2. 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.
3. Isolate one of the inverse trigonometric functions, for example, write \(\arcsin(2x) = \frac{\pi}{6} - \arccos(x)\).
4. 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 \(2x = \sin\left(\frac{\pi}{6} - \arccos(x)\right)\).
5. 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.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin and arccos, return the angle whose sine or cosine is a given value. They are essential for solving equations where the variable is inside a trigonometric function, allowing us to isolate and find angle measures.

Domain and Range Restrictions

Each inverse trig function has specific domain and range limits; for example, arcsin(x) is defined for x in [-1,1] with range [-π/2, π/2], and arccos(x) has the same domain but range [0, π]. Understanding these restrictions is crucial to find valid solutions and avoid extraneous answers.

Trigonometric Identities and Equation Manipulation

Using identities like arcsin a + arccos a = π/2 helps simplify expressions and solve equations. Manipulating the given equation by applying such identities or expressing one inverse function in terms of the other aids in isolating the variable and finding exact solutions.