Solve for x sin(x) = square root of 3-sin(x)

Problem

sin(x)=√(,3−sin(x))

Solution

1. Square both sides of the equation to eliminate the radical, noting that sin(x) must be non-negative because it equals a square root.

sin2(x)=3−sin(x)

2. Rearrange the equation into a standard quadratic form by moving all terms to one side.

sin2(x)+sin(x)−3=0

3. Apply the quadratic formula u=(−b±√(,b2−4*a*c))/(2*a) where u=sin(x) a=1 b=1 and c=−3

sin(x)=(−1±√(,1−4*(1)*(−3)))/(2*(1))

4. Simplify the expression under the radical.

sin(x)=(−1±√(,13))/2

5. Evaluate the range of the sine function, which is [−1,1] We calculate the approximate values of the roots.

(−1+√(,13))/2≈1.3028

(−1−√(,13))/2≈−2.3028

6. Determine the validity of the solutions. Since both values are outside the interval [−1,1] there is no real value of x that satisfies the equation.

sin(x)≠(−1±√(,13))/2

Final Answer

sin(x)=√(,3−sin(x))⇒No real solution

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