Solve : cosxtanx + root3cosx - root3/2tanx - 3/2 = 0
Give exact answers and formula for the general solution as well as the specific solutions..
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To solve the equation cos(x)tan(x) + √3cos(x) - (√3/2)tan(x) - 3/2 = 0, we need to simplify the equation and use trigonometric identities.
Let's simplify the equation step by step:
1. Move all the terms to one side:
cos(x)tan(x) + √3cos(x) - (√3/2)tan(x) - 3/2 = 0
Subtract (√3/2)tan(x) from both sides:
cos(x)tan(x) + √3cos(x) - 3/2 - (√3/2)tan(x) = 0
Combine the terms involving tangents:
(cos(x)tan(x) - (√3/2)tan(x)) + √3cos(x) - 3/2 = 0
Factor out tan(x):
tan(x)(cos(x) - (√3/2)) + √3cos(x) - 3/2 = 0
2. Use the trigonometric identity tan(x) = sin(x)/cos(x) to rewrite the equation:
(sin(x)/cos(x))(cos(x) - (√3/2)) + √3cos(x) - 3/2 = 0
Cancel out the common factors of cos(x):
sin(x) - (√3/2)sin(x) + √3cos(x) - 3/2 = 0
3. Rearrange the terms:
(√3cos(x) - (√3/2)sin(x)) + sin(x) - 3/2 = 0
4. Use the trigonometric identity √3cos(x) - (√3/2)sin(x) = sin(x + π/3) to rewrite the equation:
sin(x + π/3) + sin(x) - 3/2 = 0
Now, let's solve the equation sin(x + π/3) + sin(x) - 3/2 = 0:
To do this, we need to use the sum-to-product identity for sine, which states that sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2). We will apply this identity to our equation:
sin(x + π/3) + sin(x) = 2sin((x + π/3 + x)/2)cos((x + π/3 - x)/2)
= 2sin((2x + π/3)/2)cos(π/3/2)
= 2sin(x + π/6)cos(π/6)
= sqrt(3)sin(x + π/6)
Therefore, our equation simplifies to:
sqrt(3)sin(x + π/6) - 3/2 = 0
Now we can solve for sin(x + π/6):
sqrt(3)sin(x + π/6) = 3/2
sin(x + π/6) = (3/2) / sqrt(3)
sin(x + π/6) = sqrt(3)/2
Using the unit circle or trigonometric table, we find that sin(π/3) = sqrt(3)/2
Therefore, we can write:
x + π/6 = π/3 + 2nπ or x + π/6 = 2π/3 + 2nπ
Simplifying these equations, we have:
x = π/3 - π/6 + 2nπ or x = 2π/3 - π/6 + 2nπ
For the general solution, where n is an integer:
x = π/6 + 2nπ or x = 5π/6 + 2nπ