sin2x = (radical 3)/ 2
solve each equation on the interval [0,2pi)
among the repertoire of "trig things I should know" has to be the trig ratios of standard angles like 0, 30, 45, 60, and 90°
one is that sin 60° = +√3/2
as well as sin 120° = √3/2 , by the CAST rule
so if sin 2x = √3/2, then
2x = 60° or 2x = 120°
x = 30° or x = 60°
secondly, since the period of sin 2x = 180°, adding 180° to any solution yields another solution , so
x = 30+180 or 210°
x = 60+180° = 240°
in radians that would be
x = π/6 , π/3 , 7π/6 and 4π/3
so what about
tan3x= radical3 / 3
I would do it the same way, except √3/3 is not associated with one of the standard angles.
did you mean √3/2 again ?
so if sin 3x = √3/2
3x = 60° or 3x = 120°
x = 20° or x = 40°
period = 360/3 = 120°
so add 120° to any angle until you go over 360°
convert all answers to radians.
e.g. 20° = π/9 radians etc
To solve the equation sin(2x) = √3/2 on the interval [0, 2π), we can use the inverse of the sine function to find the values of x that satisfy the equation.
Step 1: Rewrite the equation
sin(2x) = √3/2
Step 2: Find the reference angle
We know that the sine function is positive (√3/2) in both the first and second quadrants. The reference angle with a sine of √3/2 is π/3.
Step 3: Set up the equation
Since the sine function has a period of 2π, we can set up the equation 2x = π/3 + 2πn and solve for x, where n is an integer.
Step 4: Solve for x
To find the values of x on the interval [0, 2π), we need to find all the solutions that satisfy 0 ≤ x < 2π.
Let's solve the equation:
2x = π/3 + 2πn
For n = 0:
2x = π/3
x = π/6
For n = 1:
2x = π/3 + 2π
2x = π/3 + 6π/3
2x = 7π/3
x = 7π/6
Since x cannot be greater than 2π, we stop here.
Therefore, the solutions to the equation sin(2x) = √3/2 on the interval [0, 2π) are x = π/6 and x = 7π/6.