Solve the trig equation exactly in the interval 0≤x≤2π.
tan x + tan π/3
------------------ = square root 3
1 - tan x tan π/3
you should recognize that they are using the formula
tan(A+B) = (tanA + tanB)/(1 - tanAtanB)
so
(tanx + tan π/3)/(1 - tanxtan(π/3) = √3
becomes
tan(x+π/3) = √3 ,
I know tan 60° or tan 240° = √3 OR tan π/3 or tan 4π/3 = √3
so x+π/3 = π/3 OR x+π/3 = 4π/3
x = 0 or x = π
but the period of tanx is π
so
x = 0 , π , 2π
To solve the trigonometric equation, we need to simplify it first. Let's rewrite the equation with a common denominator:
(tan x + tan π/3) / (1 - tan x tan π/3) = √3
Multiplying both sides of the equation by (1 - tan x tan π/3) yields:
(tan x + tan π/3) = √3(1 - tan x tan π/3)
Expanding the right side of the equation:
tan x + tan π/3 = √3 - √3 tan x tan π/3
Now, let's use a trigonometric identity to simplify the equation. The identity we can use is:
tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
By comparing the right side of the equation to the identity, we can see that A = x and B = π/3, so we can rewrite the equation as:
tan(x + π/3) = √3
Taking the inverse tangent (arctan) of both sides of the equation:
x + π/3 = arctan(√3)
To isolate x, we subtract π/3 from both sides of the equation:
x = arctan(√3) - π/3
Now we have the exact solution for x. To find the solution in the given interval 0 ≤ x ≤ 2π, we need to check if the solution falls within that interval. Let's calculate the value of x:
x = arctan(√3) - π/3 ≈ 1.047 - 1.047 ≈ 0
The solution x = 0 falls within the interval 0 ≤ x ≤ 2π. Therefore, the exact solution to the trigonometric equation in that interval is x = 0.