sin(cos^-1(x/2))=
draw right triangle to simplify
hypotenuse = 2
adjacent = x
so
opposite = sqrt(4 - x^2)
sin = sqrt(4-x^2) / 2
To simplify the expression sin(cos⁻¹(x/2)), we can make use of a right triangle.
Let's start by assuming an angle θ such that cos⁻¹(x/2) = θ. This means that the cosine of θ is equal to x/2.
We can now draw a right triangle where the adjacent side is x/2 and the hypotenuse is 1 (since the cosine of an angle is defined as the adjacent over hypotenuse in a right triangle).
Let's label the opposite side of the triangle as y.
Using the Pythagorean theorem, we know that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. So, we have:
(x/2)² + y² = 1²
(x²/4) + y² = 1
x² + 4y² = 4
Now, let's focus on the function sin(θ). In a right triangle, the sin of an angle is defined as the opposite side over the hypotenuse. Therefore:
sin(θ) = y/1 = y
So, sin(θ) = y.
Putting it all together, we have:
sin(cos⁻¹(x/2)) = sin(θ) = y
Therefore, the simplified expression of sin(cos⁻¹(x/2)) is simply y, which represents the opposite side of the right triangle we created.
Note that this only simplifies the expression, and if you have a specific value for x, you can substitute it back into the equation to find the exact value of sin(cos⁻¹(x/2)).