Solve the equation in the interval [0,2pi]. List all solutions
tan^2(x)+0.8 tan(x)-3.84 = 0
let z = tan x
z^2 +.8 z -3.84 = 0
z = [ -.8 +/- sqrt (.64 +15.36)]/2
z = [ -.8 +/- sqrt (16) ] / 2
z = [ -.8 +/- 4 ]/2
z = tan x = -2.4 or -1.6
tan is negative in quadrants 2 and 4
so
about -1 radian or about -1.18 radian
which is 2 pi - 1 and 2 pi - 1.18
also pi -1 and pi - 1.18
To solve the equation tan^2(x) + 0.8 tan(x) - 3.84 = 0 in the interval [0, 2pi], we can use a combination of algebraic manipulation and trigonometric identities. Here's how you can solve it step by step:
Step 1: Recognize the quadratic equation
The equation given is a quadratic equation in terms of tan(x). So, let's define tan(x) as a variable, say u. Now rewrite the equation using u instead of tan(x):
u^2 + 0.8u - 3.84 = 0
Step 2: Solve the quadratic equation
To solve this quadratic equation, we can use the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 0.8, and c = -3.84. Plugging these values into the quadratic formula and simplifying, we get:
u = (-0.8 ± √(0.8^2 - 4(1)(-3.84))) / (2(1))
u = (-0.8 ± √(0.64 + 15.36)) / 2
u = (-0.8 ± √16) / 2
u = (-0.8 ± 4) / 2
Now, we have two possible values for u:
u1 = (-0.8 + 4) / 2 = 3.2 / 2 = 1.6
u2 = (-0.8 - 4) / 2 = -4.8 / 2 = -2.4
Step 3: Relate u to tan(x)
Now, substitute u back with tan(x) to relate it to the original equation:
For u1: tan(x) = 1.6
For u2: tan(x) = -2.4
Step 4: Find the angles
To find the angles within the given interval [0, 2pi] that satisfy the equations tan(x) = 1.6 and tan(x) = -2.4, we can use the inverse tangent function (tan^-1) on both sides. Applying the inverse tangent function to both equations, we get:
For tan(x) = 1.6, x = tan^-1(1.6) ≈ 0.9828 radians, ≈ 56.31 degrees
For tan(x) = -2.4, x = tan^-1(-2.4) ≈ -1.1914 radians, ≈ -68.36 degrees
Step 5: Determine the solutions within the interval [0, 2pi]
In the interval [0, 2pi], we need to find the solutions that fall within this range. Let's convert the angles to the desired range:
For x ≈ 0.9828 radians, within the range [0, 2pi], the solution is x ≈ 0.9828 radians.
For x ≈ -1.1914 radians, within the range [0, 2pi], add 2pi to get a positive angle:
x ≈ -1.1914 + 2pi ≈ 5.0919 radians.
Therefore, the solutions to the equation tan^2(x) + 0.8 tan(x) - 3.84 = 0 in the interval [0, 2pi] are:
x ≈ 0.9828 radians
x ≈ 5.0919 radians