Find the smallest positive value of x(in degree) for which tan(x+100°)=tan(x+50°)*tanx*tan(x-50°)
Try an approximate graphical solution and refine solution by iterations.
Here's a graphical solution (x is in radians)
http://imageshack.us/photo/my-images/853/1313249887.png/
To find the smallest positive value of x (in degrees) for which the given trigonometric equation holds true, we can break down the problem into steps. Here's how to approach it:
Step 1: Simplify the equation
Using the trigonometric identity: tan(A + B) = (tanA + tanB) / (1 - tanA * tanB), we can rewrite the equation as follows:
(tan(x) + tan(100°)) / (1 - tan(x) * tan(100°)) = (tan(x) + tan(50°)) * tan(x) * (tan(x - 50°))
Step 2: Eliminate the denominators
Since the equation involves fractions, we can eliminate the denominators by multiplying both sides of the equation by the product of the denominators. Doing this, the equation becomes:
(tan(x) + tan(100°)) * (1 - tan(x) * tan(100°)) = (tan(x) + tan(50°)) * tan(x) * (tan(x - 50°)) * (1 - tan(x) * tan(100°))
Step 3: Expand and simplify
Multiply out both sides of the equation and collect like terms. After simplification, the equation now becomes:
tan(x) - tan(x)^3 * tan(100°) + tan(100°) + tan(100°) * tan(x)^2 = tan(x)^3 * tan(x - 50°) + tan(x - 50°) * tan(x)^2 + tan(x) * tan(x - 50°) + tan(x - 50°)
Step 4: Rearrange the equation
Rearrange the equation so that all terms are on one side and the equation is set equal to zero:
tan(x)^3 * tan(100°) + tan(100°) * tan(x)^2 + tan(x) * tan(x - 50°) + tan(x - 50°) - tan(x) + tan(x)^3 * tan(x - 50°) + tan(x - 50°) * tan(x)^2 = 0
Step 5: Apply factoring
Factor out the common factor tan(x):
tan(x) * (tan(x)^2 * tan(100°) + tan(100°) * tan(x) + tan(x - 50°) + 1) + tan(x)^2 * tan(x - 50°) + tan(x - 50°) * tan(x)^2 = 0
Step 6: Solve for possible values of x
Set each factor equal to zero and solve for the possible values of x:
tan(x) = 0
tan(x)^2 * tan(100°) + tan(100°) * tan(x) + tan(x - 50°) + 1 = 0
tan(x)^2 * tan(x - 50°) + tan(x - 50°) * tan(x)^2 = 0
Step 7: Solve each equation
Solve each equation separately to find the possible values of x. By solving these equations, we can find the smallest positive value of x for which the given trigonometric equation holds true.