solve the equation:
arccosx+arccos2x=arccos1/2
To solve the equation arccos(x) + arccos(2x) = arccos(1/2), we will use the properties of the inverse cosine function and trigonometric identities.
Step 1: Start by rewriting the equation using the cosine function to remove the inverse cosine: cos(arccos(x)) + cos(arccos(2x)) = cos(arccos(1/2))
Simplifying, we get: x + 2x = 1/2
Step 2: Combine like terms on the left side of the equation: 3x = 1/2
Step 3: Divide both sides of the equation by 3 to isolate x: x = (1/2) / 3
Step 4: Simplify: x = 1/6
Therefore, the solution to the equation arccos(x) + arccos(2x) = arccos(1/2) is x = 1/6.
It is important to note that when solving inverse trigonometric equations, it is crucial to consider the principal values or restrictions of the inverse trigonometric functions in order to find all possible solutions. Additionally, double-checking the solution obtained by plugging x = 1/6 back into the original equation is recommended to ensure its accuracy.