Given that theta is an obtuse angle measured in radians and that sin theta = k, in terms of k, an expression for cos theta?

θ is an obtuse angle, i.e. π/2 < θ < π
Hence, sinθ = k > 0  and  cosθ < 0

sin²θ + cos²θ = 1
k² + cos²θ = 1
cos²θ = 1 - k²
cosθ = -√(1 - k²)

Consider the example sin(30) = sin(150) = 1/2
cos (30) = √(3)/2 but cos(150) = -√(3)/2
Cosines are negative in the second quadrant

In general (cos θ)^2 = 1 – k^2
with positive or negative options for cos θ
The obtuse angle means θ is in the second quadrant
and that means cos θ is the negative quantity
cos θ = -√(1 – k^2)

The Pythagorean identity
-sqrt(1 - k^2)
Negative because obtuse