hi im back..
solve cosx= (squareroot3)/ 2 where 0 <(with a line underneath) x <(with a line underneath) 2 pie
a) how many solutions are possible?
b) in which quadrants would you find the solutions?
c) determine the related angle for the equation.
d) determine all the solutions for the equation.
YOUR HELP IS APPRECIATED HIGHLY <3
Hello again! I'd be happy to help you with your questions about solving the equation cos(x) = √3/2 (where 0 < x < 2π).
a) To determine the number of solutions, we need to think about the behavior of the cosine function. The cosine function has a period of 2π, which means it repeats every 2π radians or 360 degrees. Since we are looking for solutions within the interval 0 < x < 2π, we are only interested in the first full revolution of the cosine function. In this interval, the cosine function intersects the value √3/2 twice. Therefore, there are two possible solutions.
b) To determine the quadrants where the solutions lie, we need to understand the sign of the cosine function in each quadrant of the unit circle. In the first quadrant (0 < x < π/2), the cosine function is positive. In the second quadrant (π/2 < x < π), the cosine function is negative. Since the value of √3/2 is positive, the solutions will be in the first and the fourth quadrants.
c) To find the related angle for the equation, we can use the inverse cosine function (or arccosine). The related angle is the angle whose cosine is equal to √3/2. Therefore, we can use the arccosine function to find this angle. Taking the arccosine (√3/2) ≈ 30° or π/6 rad. This is the related angle for the equation.
d) To determine all the solutions, we can start with the related angle we found (π/6 rad) and consider its reflection across the x-axis to find the second solution. In the first quadrant, the angle is π/6 rad, and in the fourth quadrant (reflected over the x-axis), the angle is 2π - π/6 = 11π/6 rad. So the two solutions for the equation cos(x) = √3/2 within the given interval are x = π/6 and x = 11π/6.