Solve the equation in the given Domain:
1-cos theta/ sin theta = sqr3, 0degrees <or equal to theta < 369 degrees
cos4x + cos 2x =0, 0<or equal to x < pi
Recall that
1-cos theta/ sin theta = tan(theta/2)
so, we want tan(theta/2) = sqrt(3)
theta/2 = 60
theta = 120
Note that theta/2 = 240 also works, but that requires theta=480, outside the domain given.
___________________________
2cos^2(2x) - 1 + cos(2x) = 0
(2cos2x - 1)(cos2x + 1) = 0
cos2x = 1/2, so 2x = 60,300, so x=30,150
cos2x = -1, so 2x = 180, so x=90
To solve the first equation, 1 - cos(theta)/sin(theta) = √3, in the given domain, we can follow these steps:
Step 1: Simplify the equation
First, multiply both sides of the equation by sin(theta) to eliminate the denominator. This gives us:
sin(theta) - cos(theta) = √3 * sin(theta)
Step 2: Rewrite the equation using trigonometric identities
We can rewrite the left side of the equation using the identity sin^2(theta) + cos^2(theta) = 1:
sin(theta) - √(1 - sin^2(theta)) = √3 * sin(theta)
Step 3: Isolate the square root term
Move the square root term to one side of the equation and square both sides to get rid of the square root:
(sin(theta) - √(1 - sin^2(theta)))^2 = (√3 * sin(theta))^2
Expanding the left side:
sin^2(theta) - 2sin(theta) * √(1 - sin^2(theta)) + 1 - sin^2(theta) = 3 * sin^2(theta)
Simplifying further:
-2sin(theta) * √(1 - sin^2(theta)) + 1 = 2sin^2(theta)
Step 4: Rearrange the terms
Move the sin^2(theta) terms to one side of the equation and simplify:
2sin^2(theta) + 2sin(theta) * √(1 - sin^2(theta)) - 1 = 0
Step 5: Solve the quadratic equation
Now, we can treat sin(theta) as a variable and solve the quadratic equation. Let's substitute sin(theta) with a variable, such as "u":
2u^2 + 2u * √(1 - u^2) - 1 = 0
This equation can be solved using various techniques, such as factoring, quadratic formula, or graphing. The solutions for u will correspond to the possible values for sin(theta).
To solve the second equation, cos(4x) + cos(2x) = 0, in the domain 0 ≤ x < π, we can follow these steps:
Step 1: Rewrite using trigonometric identities
We can rewrite the equation using the identity:
cos(2x) = 2cos^2(x) - 1
This gives us:
cos(4x) + 2cos^2(x) - 1 = 0
Step 2: Simplify
Rearrange the terms:
2cos^2(x) + cos(4x) - 1 = 0
Step 3: Factor or use the quadratic formula
We can now attempt to factor this expression or use the quadratic formula to solve for cos(x). Once we have the solutions, we can check which values fall within the given domain.
Note: If the equation cannot be factored easily, it is recommended to use the quadratic formula:
cos(x) = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 1, and c = -1. Substituting these values into the quadratic formula, we can find the values of cos(x) and then check which ones satisfy the given domain.