Explanation:
To solve the equation sec(x) cos(3x) = 0 from -π/2 to π/2, we can start by considering the two factors:
1. sec(x) = 0
2. cos(3x) = 0
1. For sec(x) = 0, we know that sec(x) = 1/cos(x). Therefore, when sec(x) = 0, cos(x) must be equal to zero. However, we need to find the values of x that satisfy this condition within the given interval.
Looking at the values of cos(x) within the given interval (-π/2 to π/2), we find that cos(x) = 0 for x = -π/2 and x = π/2.
2. For cos(3x) = 0, we need to find the values of x that make cos(3x) equal to zero within the given interval.
We know that cos(3x) = 0 when 3x = π/2 + kπ/2 or 3x = 3π/2 + kπ/2, where k is an integer.
Solving for x in the first equation, we have:
3x = π/2 + kπ/2
x = (π/2 + kπ/2) / 3
Solving for x in the second equation, we have:
3x = 3π/2 + kπ/2
x = (3π/2 + kπ/2) / 3
We need to find these values of x within the given interval. Plugging in k = 0, 1, -1, 2, -2, we get the following values:
For k = 0:
x = (π/2) / 3 = π/6
For k = 1:
x = (π/2 + π/2) / 3 = π/3
For k = -1:
x = (π/2 - π/2) / 3 = 0
For k = 2:
x = (π/2 + 2π/2) / 3 = 5π/6
For k = -2:
x = (π/2 - 2π/2) / 3 = -π/6
Therefore, the solutions for cos(3x) = 0 within the interval -π/2 to π/2 are x = 0, π/6, π/3, 5π/6, and -π/6.
Now, we need to consider the solutions for sec(x) = 0 and cos(3x) = 0 together.
The solutions that satisfy both conditions are x = -π/2, π/2, and π/6.
Hence, the solutions to the equation sec(x) cos(3x) = 0 from -π/2 to π/2 are x = -π/2, π/6, π/2.