solve equation for the exact solutions if possible put answer in degrees.
2tanx/3-tan^2x=1
I have a suspicion that you meant
2tanx/(3 - tan^2x) = 1
then
2tanx = 3 - tan^2x
tan^2x + 2tanx - 3 = 0
(tanx + 3)(tanx - 1) = 0
tanx = -3 or tanx = 1
if tanx = -3, x must be in II or IV by the CAST rule
x = 108.4° or 288.4°
if x = 1, then x must be in I or III
x = 45° or x = 225°
To solve the equation 2tan(x)/3 - tan^2(x) = 1, we can rearrange the equation to get a quadratic equation in terms of tan(x):
tan^2(x) + 2tan(x)/3 - 1 = 0
Let's introduce a substitution to simplify the equation. Let u = tan(x). Applying this substitution, we have:
u^2 + 2u/3 - 1 = 0
To solve this quadratic equation, we can use the quadratic formula:
u = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = 2/3, and c = -1. Plugging these values into the quadratic formula, we get:
u = (-2/3 ± √((2/3)^2 - 4(1)(-1)))/(2(1))
u = (-2/3 ± √(4/9 + 4))/(2)
u = (-2 ± √(4 + 4))/(6)
u = (-2 ± √(8))/(6)
u = (-2 ± 2√2)/(6)
Simplifying further, we have:
u = (-1 ± √2)/(3)
Since we set u = tan(x), we can now solve for x using the inverse tangent function:
x = arctan((-1 ± √2)/(3))
The exact solutions in degrees are:
x = arctan((-1 + √2)/(3)), arctan((-1 - √2)/(3))
Please note that the ± symbol indicates that there are two possible solutions for x.
To solve the equation 2tan(x)/3 - tan^2(x) = 1 for the exact solutions in degrees, we will follow these steps:
Step 1: Simplify the equation as much as possible.
Step 2: Rearrange the equation to a quadratic equation form.
Step 3: Solve the quadratic equation.
Step 4: Find the exact solutions in degrees.
Let's start with step 1:
Step 1: Simplify the equation as much as possible.
First, we can simplify 2tan(x)/3 by multiplying the numerator and denominator by 3 to get (6tan(x))/3 = 2tan(x).
Now, our equation becomes: 2tan(x) - tan^2(x) = 1.
Moving on to step 2:
Step 2: Rearrange the equation to a quadratic equation form.
Let's move everything to one side of the equation:
tan^2(x) - 2tan(x) + 1 = 0.
This quadratic equation can be factored as (tan(x) - 1)^2 = 0.
Next, we move to step 3:
Step 3: Solve the quadratic equation.
Since (tan(x) - 1)^2 = 0, we have two cases to consider:
Case 1: tan(x) - 1 = 0
Solving this equation, we get tan(x) = 1. Taking the inverse tangent (arctan) of both sides gives us x = arctan(1), which is x = 45 degrees.
Case 2: tan(x) - 1 = 0
Solving this equation, we get tan(x) = -1. Taking the inverse tangent (arctan) of both sides gives us x = arctan(-1), which is x = -45 degrees.
Now, let's move to step 4:
Step 4: Find the exact solutions in degrees.
From our cases, we found that x = 45 degrees and x = -45 degrees are the solutions to the equation 2tan(x)/3 - tan^2(x) = 1.