Finding Solutions to the Equation
To solve the equation sec²(x) = 2tan²(x) within the interval [0, 2π], we start by rewriting the terms using the definitions of secant and tangent:
- sec(x) = 1/cos(x)
- tan(x) = sin(x)/cos(x)
This allows us to express the equation in terms of sine and cosine:
sec²(x) = 1/cos²(x) and tan²(x) = sin²(x)/cos²(x). Substituting these into the equation gives us:
1/cos²(x) = 2(sin²(x)/cos²(x))
Multiplying both sides by cos²(x) (and noting that cos(x) cannot be zero), we simplify to:
1 = 2sin²(x)
Rearranging leads to:
2sin²(x) = 1, or sin²(x) = 1/2.
Taking the square root of both sides, we find:
sin(x) = ±√(1/2) = ±√2/2.
Next, we identify the angles where the sine function equals √2/2 and -√2/2:
1. Solutions for sin(x) = √2/2
This is true at:
- x = π/4
- x = 3π/4
2. Solutions for sin(x) = -√2/2
This is true at:
- x = 5π/4
- x = 7π/4
Final Solutions
Combining all the solutions, we find the values of x within the interval [0, 2π]:
- x = π/4
- x = 3π/4
- x = 5π/4
- x = 7π/4
These angles represent the complete set of solutions to the equation sec²(x) = 2tan²(x) in the specified interval.