Solve sec to the power of 2 times x minus 1 = 3, x such that [0,2pi).
By "sec" do you mean 2?
if you mean
sec^2 x - 1 = 3
then we can proceed to
sec^2 x = 4
sec x = 2 or -2
sec pi/3 = 2, so using that as a reference angle in each quadrant,
x = pi/3 or 2pi/3 or 4pi/3 or 5pi/3
To solve the equation sec^2(x) - 1 = 3, we need to isolate the variable x. Here's how you can solve it step by step:
1. Start with the initial equation: sec^2(x) - 1 = 3.
2. Add 1 to both sides of the equation: sec^2(x) = 4.
3. Take the square root of both sides to eliminate the square: sqrt(sec^2(x)) = sqrt(4).
4. Remember that sec^2(x) is the square of the secant function, so taking the square root gives us the positive and negative values of the secant function: sec(x) = ±2.
5. Secant function is the reciprocal of cosine function, so we can rewrite sec(x) = ±2 as 1/cos(x) = ±2.
6. Flip both sides of the equation to get rid of the fraction: cos(x) = ±1/2.
7. Solve for x by finding the inverse cosine of both sides: x = arccos(±1/2).
8. The solutions for x can be found using the unit circle or a calculator. On the unit circle, cos(x) = ±1/2 occurs at specific angles. In this case, cos(x) = 1/2 occurs at pi/3 and 5pi/3, and cos(x) = -1/2 occurs at 2pi/3 and 4pi/3.
9. Since the given range for x is [0, 2pi), we need to check which solutions fall within this range.
10. The values pi/3 and 5pi/3 fall within the range, so these are valid solutions.
Therefore, the solutions for x in the given range [0, 2pi) are x = pi/3 and x = 5pi/3.