sin θ=2/3 and θ correspond to a point in quadrant 2. cos θ=
sinθ = y/r
So, in QII,
y = 2
r = 3
x = -√5
Now, cosθ = x/r = ?
To find the value of cos θ, we can use the Pythagorean identity.
Since sin θ = 2/3 and θ corresponds to a point in quadrant 2, we know that sin θ is positive in quadrant 2.
Using the Pythagorean identity, we have:
cos^2 θ + sin^2 θ = 1
Plugging in the value of sin θ, we get:
cos^2 θ + (2/3)^2 = 1
Simplifying, we have:
cos^2 θ + 4/9 = 1
cos^2 θ = 1 - 4/9
cos^2 θ = 5/9
Taking the square root of both sides, we find:
cos θ = ±√(5/9)
Since θ is in quadrant 2, the cosine function is negative in this quadrant. Therefore, cos θ = -√(5/9).
So, cos θ = -√(5/9).
To find the value of cos θ, we can use the Pythagorean Identity formula, which states that sin^2 θ + cos^2 θ = 1.
Given that sin θ = 2/3, we can square both sides to get (sin θ)^2 = (2/3)^2. This simplifies to sin^2 θ = 4/9.
Now, we can substitute this value into the Pythagorean Identity formula: sin^2 θ + cos^2 θ = 1. Plugging in sin^2 θ = 4/9, we get:
4/9 + cos^2 θ = 1
To solve for cos^2 θ, we subtract 4/9 from both sides:
cos^2 θ = 1 - 4/9
Simplifying the right side gives:
cos^2 θ = 9/9 - 4/9
cos^2 θ = 5/9
Taking the square root of both sides, we find:
cos θ = ± √(5/9)
Since θ corresponds to a point in quadrant 2, which is where cosine is negative, the value of cos θ is negative. Therefore:
cos θ = -√(5/9)
Final Answer: cos θ = -√(5/9)