solve 0=(3^0.5)tan(x+(pi/6))+1 and find all general solutions
(3^0.5)tan(x+(pi/6))+1 = 0
I am going to let x=pi/6 = y for easier typing
(3^0.5)tan(y)+1 = 0
tany = -1/√3 ...... ah!, tan 30º = +1/√3
so y must be in the second or fourth quadrants.
y = 150º or y = 330º or (5pi/6, 11pi/6)
then x+pi/6 = 5pi/6
x = 2pi/3
or
x+pi/6 = 11pi/6
x = 5pi/3
the period of your function is pi
so the general solutions:
x = 2pi/3 + kpi or
x = 5pi/3 + kpi , where k is an integer.
To solve the equation 0 = √3 tan(x + π/6) + 1 and find all general solutions, we can follow these steps:
Step 1: Subtract 1 from both sides of the equation to isolate the square root term:
√3 tan(x + π/6) = -1
Step 2: Divide both sides of the equation by √3 to eliminate the square root term:
tan(x + π/6) = -1/√3
or
tan(x + π/6) = -√3/3
Step 3: Convert the tangent to sine and cosine using the identity tan(theta) = sin(theta)/cos(theta):
sin(x + π/6) / cos(x + π/6) = -√3/3
Step 4: Use the sum-to-product trigonometric identity to simplify the equation further:
(sin(x)cos(π/6) + cos(x)sin(π/6)) / (cos(x)cos(π/6) - sin(x)sin(π/6)) = -√3/3
Simplifying this equation further:
(sin(x) + √3cos(x)) / (cos(x) - √3sin(x)) = -√3/3
Step 5: Cross-multiply and simplify to get rid of fractions:
(sin(x) + √3cos(x))(-√3sin(x) - 3cos(x)) = -3(cos(x) - √3sin(x))
Expanding and simplifying:
-3sin(x)sin(x) - 3√3cos(x)sin(x) - 3√3sin(x)cos(x) - 3√3cos(x)cos(x) = -3cos(x) - 3√3sin(x)
Simplifying further:
-3sin^2(x) - 3√3sin(x)cos(x) - 3√3cos(x)sin(x) - 3√3cos^2(x) = -3cos(x) - 3√3sin(x)
Re-arranging the terms:
-3sin^2(x) - 6√3sin(x)cos(x) - 3√3cos^2(x) = -3cos(x) - 3√3sin(x)
Step 6: Move all terms to one side of the equation:
-3sin^2(x) - 6√3sin(x)cos(x) + 3cos(x) - 3√3cos^2(x) - 3√3sin(x) + 3√3sin(x) = 0
Simplifying further:
-3sin^2(x) - 6√3sin(x)cos(x) - 3√3cos^2(x) + 3cos(x) = 0
Step 7: Factor out common terms:
-3(sin^2(x) + 2√3sin(x)cos(x) + √3cos^2(x) - cos(x)) = 0
Step 8: Use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify:
-3((1 + 2√3sin(x)cos(x)) - cos(x)) = 0
Simplifying further:
-3(1 + 2√3sin(x)cos(x) - cos(x)) = 0
Step 9: Distribute the negative sign:
-3 - 6√3sin(x)cos(x) + 3cos(x) = 0
Step 10: Combine like terms:
3cos(x) - 6√3sin(x)cos(x) - 3 = 0
Step 11: Factor out cos(x):
cos(x)(3 - 6√3sin(x) - 1) = 3
Simplifying further:
cos(x)(2 - 6√3sin(x)) = 3
Step 12: Divide both sides by (2 - 6√3sin(x)):
cos(x) = 3 / (2 - 6√3sin(x))
This equation can be solved for values of x by considering the inverse cosine function.
Thus, the general solutions for the equation 0 = √3 tan(x + π/6) + 1 are obtained by finding the values of x for which cos(x) = 3 / (2 - 6√3sin(x)).