Question

Simplify the expression: sin(2cos^(-1)(4/9))

Answer 1

To simplify the expression
`sin(2cos-1(4/9))`, we use the double-angle identity and the Pythagorean identity to find the sine of the angle given by the inverse cosine of
`4/9.`

Step-by-step explanation:

To simplify the expression:
`sin(2cos-1(4/9))`, we can use the double-angle identity for sine and the definition of the inverse cosine function. The double-angle identity is
`sin(2α) = 2sin(α)cos(α)`. First, we recognize that
`cos-1(4/9)` is the angle whose cosine is
`4/9`. Let's call this angle
`θ`, so we have
`θ = cos-1(4/9)`, and therefore
`cos(θ) = 4/9`.

Since
`sin2(θ) + cos2(θ) = 1` (the Pythagorean identity), we can find
`sin(θ)` by solving
`sin2(θ) = 1 - cos2(θ) = 1 - (4/9)2`.

This gives us
`sin(θ) = √(1 - (4/9)2)`, considering only the positive root since the angle
`θ` must be in the rang
`[0, π]` for
`cos-1`.

Plugging the values of
`sin(θ)` and
`cos(θ)` into the double-angle identity, we get
`sin(2θ) = 2sin(θ)cos(θ) = 2(√(1 - (4/9)2))(4/9)` which simplifies to the final answer.