Solve cosxtanx + root3cosx - root3/2tanx - 3/2 = 0
exact answers and formula for general solution
To solve the equation cos(x)tan(x) + √3cos(x) - (√3/2)tan(x) - 3/2 = 0, we can rearrange the terms and apply some trigonometric identities to simplify the expression.
First, we can group the terms containing tan(x) and cos(x) together:
(cos(x)tan(x) - (√3/2)tan(x)) + √3cos(x) - 3/2 = 0
Next, we can factor out tan(x):
tan(x)(cos(x) - √3/2) + √3cos(x) - 3/2 = 0
We can simplify the expression √3/2 by rationalizing the denominator:
tan(x)(cos(x) - √3/2) + 2√3cos(x)/2 - 3/2 = 0
tan(x)(cos(x) - √3/2) + (2√3cos(x) - 3)/2 = 0
Next, we can multiply the entire equation by 2 to eliminate the fraction:
2tan(x)(cos(x) - √3/2) + 2(2√3cos(x) - 3)/2 = 0
simplifying further:
2tan(x)(cos(x) - √3/2) + 2√3cos(x) - 3 = 0
Now we have a quadratic equation in terms of cos(x):
2tan(x)·cos(x) - tan(x)·√3 + 2√3cos(x) - 3 = 0
To solve this equation, we can use the quadratic formula:
cos(x) = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = 2tan(x), b = -tan(x)√3 + 2√3, and c = -3.
Plugging in these values, we get:
cos(x) = [-(-tan(x)√3 + 2√3) ± √((-tan(x)√3 + 2√3)^2 - 4(2tan(x))(-3))] / (2(2tan(x)))
Simplifying this further will give you the exact solutions for cos(x).
To find the general solution, we need to find all the values of x that satisfy the equation. We can use the periodicity of trigonometric functions to find the general solution.
For example, cos(x) has a period of 2π, and tan(x) has a period of π. To find the general solution, we need to find all the angles in the interval [0, 2π] or [0, π] that satisfy the equation.
After obtaining the values of x using the quadratic formula, we can add multiples of 2π or π to these solutions to find all the possible values of x in the given interval.
Note: This solution assumes that x is measured in radians.