Solve the equation for solutions in the interval 0<=theta<2pi
Problem 1.
3cot^2-4csc=1
My attempt:
3(cos^2/sin^2)-4/sin=1
3(cos^2/sin^2) - 4sin/sin^2 = 1
3cos^2 -4sin =sin^2
3cos^2-(1-cos^2) =4sin
4cos^2 -1 =4sin
Cos^2 - sin=1/4
(1-sin^2) - sin =1/4
-Sin^2 - sin =-3/4
Sin(sin+1) =3/4
Sin= 3/4. Or sin =-1/4
I don't think these give the right answers for theta. I keep getting an error on my calculator
from
3cos^2 Ø -4sinØ =sin^2 Ø , why not replace the cos^2 Ø
3(1 - sin^2 Ø) - 4sinØ - sin^2 Ø = 0
..
4sin^2 Ø + 4sinØ - 3 = 0
(2sinØ - 1)(2sinØ + 3) = 0
sinØ = 1/2 or sinØ = -3/2 which is not possible
sinØ = 1/2
Ø = π/6 or 5π/6 , (30° or 150°)
your solution breaks down here:
Sin(sin+1) =3/4
Sin= 3/4. Or sin =-1/4
you can only set factors equal to zero if the right side of your equation is zero.
My second concern is your poor usage of trig function names.
You need an argument after the trig name
e.g. sinØ = .3
sin = .3 is meaningless, and is just that, a sin .
that's like saying √ = 2
You cannot work this:
sin(sin+1) = 3/4
if sin=3/4, sin+1 = 7/4
This is why we always set things to zero. If a product is zero, one of the factors nust be zero.
sin^2+sin = 3/4
4sin^2+4sin-3=0
(2sin-1)(2sin+3) = 0
sin = 1/2 or sin = -3/2
only sin = 1/2 is allowed, so
θ = π/6, 5π/6
or, since cot^2 = csc^2-1, you have
3(csc^2-1)-4csc - 1 = 0
3csc^2 - 4csc - 4 = 0
(3csc+2)(csc-2) = 0
csc = 2 or -2/3
|csc| is never less than 1, so
csc = 2 is the only solution.
Same as above
Let's go through the steps together to solve the equation.
Starting from your attempt:
3cot^2 - 4csc = 1
Rewrite cot and csc in terms of sin and cos:
3(cos^2/sin^2) - 4/sin = 1
Multiply both sides by sin^2 to get rid of the denominators:
3cos^2 - 4sin = sin^2
Rearrange the equation:
3cos^2 - sin^2 - 4sin = 0
Now, let's solve for sin. We can rewrite the equation as:
3cos^2 - sin^2 - 4sin = 0
Multiply through by -1 to change the sign of every term:
sin^2 + 4sin - 3cos^2 = 0
Now, let's solve this quadratic equation for sin. Factoring the equation:
sin^2 + 4sin - 3cos^2 = 0
(sin - 1)(sin + 3) = 0
So, we have two possibilities:
1) sin - 1 = 0
sin = 1
2) sin + 3 = 0
sin = -3
Now, let's check these solutions in the original equation:
When sin = 1:
3cot^2 - 4csc = 1
3(cot^2) - 4(csc) = 1
3(1/tan^2) - 4(1/sin) = 1
3(1/(sin^2/cos^2)) - 4(1/(sin)) = 1
3cos^2 - 4sin = sin^2
3(1 - sin^2) - 4sin = sin^2
-3sin^2 - 4sin + 3 = 0
Solving this quadratic equation for sin gives sin = 1. So, sin = 1 is a solution.
When sin = -3:
3cot^2 - 4csc = 1
3(cot^2) - 4(csc) = 1
3(1/tan^2) - 4(1/sin) = 1
3(1/(sin^2/cos^2)) - 4(1/(sin)) = 1
3cos^2 - 4sin = sin^2
3(1 - sin^2) - 4sin = sin^2
-3sin^2 - 4sin + 3 = 0
Solving this quadratic equation for sin gives sin = -3, but since the interval for theta is 0 <= theta < 2pi, sin cannot be -3. Therefore, sin = -3 is not a valid solution in this case.
So, the solution is sin = 1. To find the corresponding values of theta, we use the inverse sine function:
sin(theta) = 1
theta = sin^(-1)(1)
The range for theta is in the interval 0 <= theta < 2pi, so the solution is:
theta = pi/2, 3pi/2
To solve the equation 3cot^2θ - 4cscθ = 1 on the interval 0 ≤ θ < 2π, let's break down the steps:
Step 1: Rewrite the equation using trigonometric identities.
Using the identities, cot^2θ = cos^2θ/sin^2θ and cscθ = 1/sinθ, we can rewrite the equation as:
3(cos^2θ/sin^2θ) - 4(1/sinθ) = 1
Step 2: Simplify the equation.
Multiply everything by sin^2θ to eliminate the denominator:
3cos^2θ - 4sin = sin^2θ
Step 3: Rearrange the equation.
Move all the terms to one side to get a quadratic equation:
3cos^2θ - sin^2θ - 4sin = 0
Step 4: Simplify the quadratic equation.
Using the identity sin^2θ = 1 - cos^2θ, we can substitute it into the equation:
3cos^2θ - (1 - cos^2θ) - 4sin = 0
Combine like terms:
4cos^2θ + 4sin - 1 = 0
Step 5: Solve the quadratic equation.
To solve the quadratic equation 4cos^2θ + 4sin - 1 = 0, you can substitute sinθ with the identity sinθ = √(1-cos^2θ):
4cos^2θ + 4√(1-cos^2θ) - 1 = 0
Unfortunately, the expression becomes more complicated, and finding explicit solutions becomes difficult. You might be getting an error on your calculator because it's challenging to solve the equation by hand or using basic calculator functionalities. You may need to use more advanced mathematical software or numerical methods to approximate the solutions within the given interval.
Alternatively, you can try using an online equation solver or software like MATLAB or Wolfram Alpha to find the solutions more accurately.