Question
simplify sin(2 arccos(x))
Answers
GPT 3.5
We can simplify sin(2 arccos(x)) using the double angle trigonometric identity for sine:
sin(2θ) = 2sin(θ)cos(θ)
First, let's find sin(arccos(x)). We know that arccos(x) gives us the angle whose cosine is x. So, if we have a right triangle with an angle θ whose cosine is x, then the opposite side would have a length of √(1 - x^2), and the hypotenuse would have a length of 1. Therefore, sin(arccos(x)) = √(1 - x^2)/1 = √(1 - x^2).
Now, using the double angle identity, we have:
sin(2 arccos(x)) = 2sin(arccos(x))cos(arccos(x))
= 2(√(1 - x^2))(x)
= 2x√(1 - x^2)
So, sin(2 arccos(x)) simplifies to 2x√(1 - x^2).
sin(2θ) = 2sin(θ)cos(θ)
First, let's find sin(arccos(x)). We know that arccos(x) gives us the angle whose cosine is x. So, if we have a right triangle with an angle θ whose cosine is x, then the opposite side would have a length of √(1 - x^2), and the hypotenuse would have a length of 1. Therefore, sin(arccos(x)) = √(1 - x^2)/1 = √(1 - x^2).
Now, using the double angle identity, we have:
sin(2 arccos(x)) = 2sin(arccos(x))cos(arccos(x))
= 2(√(1 - x^2))(x)
= 2x√(1 - x^2)
So, sin(2 arccos(x)) simplifies to 2x√(1 - x^2).