Use substitution to determine whether the given x-value is a solution of the equation:
sin x = -2sqrt3/3, x=4pi/3
A) Yes
B) No
C) More information is needed
D) The problem is indeterminate
To determine whether the given x-value is a solution of the equation sin x = -2√3/3, we can substitute the x-value into the equation and check if both sides are equal.
Substituting x = 4π/3 into the equation sin x = -2√3/3, we have:
sin(4π/3) = -2√3/3
Now, we need to evaluate sin(4π/3). To do this, recall that sin is a periodic function with a period of 2π. This means that any angle t is congruent to t + 2πn for any integer n. Therefore, we can find an equivalent angle within the range of 0 to 2π that has the same sine value.
To find an equivalent angle, we can subtract or add multiples of 2π until we get an angle within the range of 0 to 2π. Since 4π/3 is already greater than 2π, we need to subtract 2π to bring it within the desired range.
4π/3 - 2π = 4π/3 - 6π/3 = -2π/3
Now, we can evaluate sin(-2π/3). Using the symmetry property of sine, sin(-x) = -sin(x), we have:
sin(-2π/3) = -sin(2π/3)
sin(2π/3) is a well-known value. It equals √3/2. Therefore:
sin(-2π/3) = -√3/2
Now we can compare the left and right sides of the equation:
-√3/2 = -2√3/3
Since both sides are equal, it means that x = 4π/3 is indeed a solution of the equation sin x = -2√3/3.
Therefore, the answer is:
A) Yes