Using the pythagorean identity, determine cos0 if sin0 = 7/9 and π/2 < 0 < π
The Pythagorean identity states that for any angle θ:
sin²θ + cos²θ = 1
Given that sinθ = 7/9 and θ is in the second quadrant (π/2 < θ < π), we can find cosθ using the identity. In the second quadrant, sine is positive but cosine is negative.
First, find sin²θ:
sin²θ = (7/9)²
sin²θ = 49/81
Now, use the Pythagorean identity to find cos²θ:
cos²θ = 1 - sin²θ
cos²θ = 1 - 49/81
cos²θ = 81/81 - 49/81
cos²θ = 32/81
So, cos²θ = 32/81. To find cosθ, we need to take the square root of 32/81. Since θ is in the second quadrant and cosine is negative there, we take the negative square root:
cosθ = -√(32/81)
cosθ = -√32 / √81
cosθ = -4√2 / 9
Therefore, cosθ equals -4√2 / 9 for an angle θ in the second quadrant when sinθ = 7/9.