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Find the solutions to the equation startroot 3 endroot sine (startfraction x over 2 endfraction) cosine x = 1 if 0° ≤ < 360°. check all that apply. 0° 120° 60° 180° 240°

User Iajrz
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Final answer:

The equation √3 sin(x/2) cos(x) = 1 does not have any solutions because the sine function has a maximum value of 1, and √3 is greater than 1, making the equation impossible to satisfy within the specified range of 0° to 360°.

Step-by-step explanation:

To solve the equation √3 sin(x/2) cos(x) = 1 within the interval 0° ≤ x < 360°, we can use trigonometric identities. Firstly, recognize that the expression can be transformed using the double angle identity sin(2θ) = 2sin(θ)cos(θ). Since we have a √3 factor, we divide the double angle identity by √3:

sin(2θ) / √3 = 2sin(θ)cos(θ) / √3

This simplifies to sin(2θ) / √3 = sin(x/2) cos(x), given that θ = x/2. Now, equate this to 1:

sin(2θ) / √3 = 1

Further simplify to:

sin(2θ) = √3

2θ, or x, will be an angle where sin(2θ) = √3. But since the maximum value of the sine function is 1, there are no solutions where sin(2θ) equals √3, as √3 > 1. Hence, there are no angles x within the range 0° to 360° that satisfy the original equation.

User Yug Kapoor
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