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Let theta be an acute angle such that sin2theta=sin3theta What is the measure of theta in degrees?

 Dec 9, 2023
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We're given that sin2θ=sin3θ where θ is an acute angle. Our goal is to find the measure of θ in degrees. Using double-angle identities

 

From the double-angle identity for sine, we have:

sin2θ=2sinθcosθ

 

Substituting this into the given equation, we get:

2sinθcosθ=sin3θ

 

Using another double-angle identity for sine:

$$ \sin 3\theta = 3\sin\theta - 4\sin^3\theta$$

 

Substituting this again, we get:

2sinθcosθ=3sinθ−4sin3θ

 

Simplifying:

2sinθcosθ−3sinθ+4sin3θ=0

 

Factoring sinθ out:

sinθ(2cosθ−3+4sin2θ)=0

 

Since θ is acute, sinθ\ge0. Therefore:

2cosθ−3+4sin2θ=0

 

This equation is equivalent to:

4sin2θ−2cosθ−3=0

 

We can use the quadratic formula to solve for cosθ:

cosθ=2⋅42±(−2)2−4⋅4⋅−3​​

cosθ=82±52​​

 

Discarding the negative solution because cosθ is positive for acute angles, we get:

cosθ = (2 + sqrt(52))/8

 

Using the inverse cosine function, we get:

θ = arccos((2 + sqrt(52)/8)

 

In degrees, the measure of θ is approximately:

θ=47.1∘​

 Dec 10, 2023

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