tan4theta = (radical 3) / 3
theta = 7pi/24, 13pi/24, 19pi/24
Let me know if im right.
tan4x = 1/√3
4x = π/6
x = π/24 + nπ/4
= π/24, 7π/24, 13π/24, 19π/24
Looks like you missed π/24
arctan(-rad3 / 3)= -pi/6 ?
yes, tan (-30 degrees) = -sqrt 3 / 3
but also tan (180-30) =pi-pi/6 = 5 pi/6
To verify if the given values for theta = 7pi/24, 13pi/24, and 19pi/24 satisfy the equation tan(4theta) = (√3)/3, we can substitute these values into the equation and check if both sides are equal.
First, let's compute tan(4theta) using the given values of theta.
For theta = 7π/24:
tan(4theta) = tan(4 * (7π/24)) = tan((28π)/24) = tan(7π/6)
Next, we need to simplify tan(7π/6) to see if it matches (√3)/3.
To do this, we can use the unit circle or trigonometric ratios.
The unit circle shows that at an angle of 7π/6 (210 degrees), the value of tangent is -(√3)/3. Since this is the negative of (√3)/3, it does not match.
Similarly, we can find the other values:
For theta = 13π/24:
tan(4theta) = tan(4 * (13π/24)) = tan((52π)/24) = tan((13π)/6)
Using the unit circle, we find that tan((13π)/6) = (√3)/3.
For theta = 19π/24:
tan(4theta) = tan(4 * (19π/24)) = tan((76π)/24) = tan((19π)/6)
Using the unit circle, we find that tan((19π)/6) = (√3)/3.
Therefore, only theta = 13π/24 and theta = 19π/24 satisfy the equation tan(4theta) = (√3)/3.
So, you are correct about theta = 13π/24 and theta = 19π/24, but theta = 7π/24 is incorrect.