Answer:
cos(θ) = -4/5
Step-by-step explanation:
You want cos(θ), given sin(θ) = 3/5 and sin(2θ) = -24/25.
Double angle
The double-angle identity for sine is ...
sin(2θ) = 2sin(θ)cos(θ)
Solving for cos(θ) gives ...
cos(θ) = sin(2θ)/(2sin(θ))
cos(θ) = (-24/25)/(2·3/5) = (-24/25)/(30/25) = -24/30
cos(θ) = -4/5
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Additional comment
The Pythagorean identity tells you the cosine is ...
cos(θ) = ±√(1 -sin(θ)²) = ±√(1 -9/25) = ±√(16/25) = ±4/5
The sine is positive in the first and 2nd quadrants. Since the double angle has a negative sine, θ cannot be a first quadrant angle. The cosine of a 2nd-quadrant angle is negative, hence -4/5.
