Final answer:
To find sinθ when given sin2θ = cos3θ for an acute angle, one should use various trigonometric identities to express both sides in terms of sinθ and then solve for sinθ.
Step-by-step explanation:
To solve the equation sin2θ = cos3θ where θ is an acute angle, we can use trigonometric identities. The double angle identity for sine is sin2θ = 2sinθcosθ and for cosine, we have an identity that expresses cosine in terms of sine: cos2θ = 1 - 2sin2θ. By substituting 3θ for θ in the cosine double angle identity, we have cos3θ = cos(2θ + θ) = cos2θcosθ - sin2θsinθ which simplifies further using the double angle identity. We then equate the expression derived from sin2θ to the expression derived from cos3θ and solve for sinθ.
Since sin2θ and cos3θ are equal and given that θ is an acute angle, we look for the solution within the range [0, π/2]. As the expressions become more complex, we can solve the equations either by applying algebraic manipulation or by using a unit circle to find the common angle that satisfies both equations.