Introduction to Trigonometric Equations

Trigonometric equations involve the use of trigonometric functions such as sine, cosine, and their inverse functions like arcsin and arccos. These equations can often be solved using various algebraic and trigonometric identities. In this article, we will explore the specific equation 2 arcsin x arccos 2x and how to solve it step-by-step.

Solving 2 arcsin x arccos 2x

To solve the equation 2 arcsin x arccos 2x, we can start by taking the cosine of both sides:

[cos(2 arcsin x) cos(arccos 2x) 2x]

Using the double-angle identity for cosine, we have:

[cos(2 arcsin x) 1 - 2 sin^2(arcsin x) 1 - 2x^2]

Substituting (cos(arccos 2x) 2x) into the equation, we get:

[1 - 2x^2 2x]

Bringing everything to one side of the equation, we obtain:

[2x^2 2x - 1 0]

This is a standard quadratic equation, which can be solved using the quadratic formula:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Plugging in the values (a 2), (b 2), and (c -1), we get:

[x frac{-2 pm sqrt{4 8}}{4} frac{-2 pm 2sqrt{3}}{4} frac{-1 pm sqrt{3}}{2}]

We must consider the range of (x) for both the sine and cosine functions. The sine function, (arcsin(x)), is defined for (-1 leq x leq 1). The range of the other root (frac{-1 - sqrt{3}}{2}) is outside of this interval, hence it is not a valid solution.

Therefore, the only valid solution is:

[x frac{-1 sqrt{3}}{2}]

Alternative Solution Using Trigonometric Identities

Let (y arcsin x), then (sin y x) and (arccos 2x 2y). This implies:

[cos(2y) 2x 2 sin y]

Using the double-angle identity (cos(2y) cos^2 y - sin^2 y), we get:

[cos^2 y - sin^2 y 2 sin y]

Substituting (cos^2 y 1 - sin^2 y) and simplifying, we obtain:

[1 - 2 sin^2 y 2 sin y]

Moving everything to one side:

[2x^2 - 2x - 1 0]

Using the quadratic formula again:

[x frac{-2 pm sqrt{4 8}}{4} frac{-2 pm 2sqrt{3}}{4} frac{-1 pm sqrt{3}}{2}]

Only the value (x frac{-1 sqrt{3}}{2}) lies within the interval ([-1, 1]), hence this is the valid solution.

Conclusion

Through both approaches, we find that the solution to the equation 2 arcsin x arccos 2x is:

[boxed{x frac{sqrt{3} - 1}{2}}]

This solution is derived using standard algebraic techniques and the properties of trigonometric functions.