Solve for x sin(x)=3

Problem

sin(x)=3

Solution

1. Identify the range of the real-valued sine function. For any real number x the value of sin(x) must fall within the interval [−1,1]

2. Compare the given value to the range. The equation states sin(x)=3 Since 3>1 there are no real solutions for x

3. Apply the inverse sine function to find complex solutions. We use the identity sin(x)=(e(i*x)−e(−i*x))/(2*i)

4. Substitute the value into the exponential form.

(e(i*x)−e(−i*x))/(2*i)=3

5. Rearrange into a quadratic form. Let u=e(i*x)

u−1/u=6*i

u2−6*i*u−1=0

6. Solve for u using the quadratic formula.

u=(6*i±√(,(−6*i)2−4*(1)*(−1)))/2

u=(6*i±√(,−36+4))/2

u=(6*i±√(,−32))/2

u=(6*i±4*i√(,2))/2

u=i*(3±2√(,2))

7. Solve for x using the natural logarithm. Since e(i*x)=u then i*x=ln(u)+2*k*π*i

x=−i*ln(i*(3±2√(,2)))+2*k*π

8. Simplify the expression using ln(i)=(i*π)/2

x=−i*(ln(3±2√(,2))+(i*π)/2)+2*k*π

x=π/2+2*k*π−i*ln(3±2√(,2))

Final Answer

sin(x)=3⇒x=π/2+2*k*π−i*ln(3±2√(,2))

Want more problems? Check here!