Use trigonometric identities to solve the equation csc^2x -(√3+1)cotx+(√3-1)=0 in the interval [0, 2pi]
One of the ID's you should know is
cot^ Ø + 1= csc^ Ø
then:
csc^2x -(√3+1)cotx+(√3-1)=0
cot^2 x + 1 - (√3+1)cotx + (√3-1)=0
cot^2 x - (√3+1) cotx + √3 = 0
cot^2 x - cotx - √3cotx + √3=0
cotx(cotx - 1) - √3(cotx -1) = 0
(cotx - 1)(cotx - √#) = 0
cotx = 1 or cotx = √3
case1: cotx = 1
then tanx = 1
x = 45°, 225° or π/4 , 5π/4 radians
case 2: cotx = √3
tanx = 1/√3
recognize the 1-√3-2 right-angled triangle,
x = 30° , 210° or π/6, 7π/6
x = π/6, π/4, 7π/6 , 5π/4
To solve the equation csc^2x - (√3+1)cotx + (√3-1) = 0 in the interval [0, 2π], we will use some trigonometric identities to simplify the equation.
Step 1: Simplify the equation using trigonometric identities.
Let's start by using the identities:
csc^2x = cot^2x + 1
cotx = cosx / sinx
csc^2x - (√3+1)cotx + (√3-1) = 0
(cot^2x + 1) - (√3+1)(cosx/sinx) + (√3-1) = 0
cot^2x - (√3+1)(cosx/sinx) + (√3-1) = 0
Step 2: Substitute cosx and sinx with their respective identities.
cot^2x - (√3+1)(cosx/sinx) + (√3-1) = 0
cot^2x - (√3+1)(1/tanx) + (√3-1) = 0
cot^2x - (√3+1)/tanx + (√3-1) = 0
cot^2x - (√3+1)/(sinx/cosx) + (√3-1) = 0
cot^2x - (√3+1)(cosx/sinx) + (√3-1) = 0
Step 3: Simplify further using the identity for cot^2x.
The identity cot^2x = (1/tan^2x) can be used to simplify the equation.
cot^2x - (√3+1)/(sinx/cosx) + (√3-1) = 0
(1/tan^2x) - (√3+1)(cosx/sinx) + (√3-1) = 0
Now, let's change everything to terms of sinx and cosx.
Step 4: Convert everything to terms of sinx and cosx.
Use the following identities:
tanx = sinx / cosx
cotx = cosx / sinx
[((1) / (sin^2x)) - ((√3+1)(cosx) / (sinx))] + (√3-1) = 0
Step 5: Rewrite the equation without fractions.
Multiply through by sin^2x:
1 - (√3+1)cosx + (√3-1)sin^2x = 0
Step 6: Rearrange the equation.
Bring all the terms to one side of the equation:
(√3-1)sin^2x - (√3+1)cosx + 1 = 0
This quadratic equation in terms of sinx can be further simplified and solved using algebraic techniques or factoring methods, but it does not have a simple exact solution.
You can solve this equation numerically by using a graphing calculator or software, or by using numerical methods like the Newton-Raphson method to find approximations of the solutions within the given interval [0, 2π].