Question

Which of the following is a solution to sin(x/2) = radical 3/2

Answer 1

To find the solution to sin(x/2) = √3/2, use the inverse sine function and multiply the result by 2.

Step-by-step explanation:

To find the solution to sin(x/2) = √3/2, we can use the inverse sine function. The inverse of sin(x/2) is sin^(-1)(x/2) or arcsin(x/2). So, we have arcsin(√3/2) = x/2. To solve for x, we can multiply both sides by 2: x = 2arcsin(√3/2).

The value of √3/2 represents the sine of an angle. In this case, it represents the sine of 60 degrees or π/3 radians. So, x = 2arcsin(π/3).

The solution to the equation sin(x/2) = √3/2 is x = 2arcsin(π/3) or x ≈ 120.96 degrees or x ≈ 2.11 radians.

Answer 2

sin ( x/ 2 ) = - √ 3 /2

Take the inverse sine of both sides of the equation to extract x

from inside the sine.

x/ 2 = arcsin ( − √ 3/ 2 )

The exact value of arcsin ( − √ 3 /2 ) is − π /3 .

/x 2 = − π /3

Multiply both sides of the equation by 2 .

2 ⋅ x /2 = 2 ⋅ ( − π /3 )

Simplify both sides of the equation.

x = − 2 π /3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from 2 π , to find a reference angle. Next, add this reference angle to π to find the solution in the third quadrant.

x /2 = 2 π + π/ 3 + π

Simplify the expression to find the second solution.

x = 2 π /3

4 π

Add 4 π to every negative angle to get positive angles.

x = 10 π /3

The period of the sin ( x /2 ) function is 4 π so values will repeat every 4 π radians in both directions.

x =2 π /3 + 4 π n , 10 π/ 3 + 4 π n , for any integer n

Exclude the solutions that do not make sin ( x /2 ) = − √ 3/ 2 true.

x = 10 π /3 + 4 π n , for any integer n