Solve: (\cos x = \sin^2 x).
Step 1: Replace (\sin^2 x = 1 - \cos^2 x).
So (\cos x = 1 - \cos^2 x).
Step 2: Rearrange: (\cos^2 x + \cos x - 1 = 0). Let (u = \cos x): (u^2 + u - 1 = 0). Solve: (\cos x = \sin^2 x)
Step 3: (u = \frac-1 \pm \sqrt1 + 42 = \frac-1 \pm \sqrt52).
Step 4: (\sqrt5 \approx 2.236), so:
Step 5: (\cos x = 0.618) → principal (x = \arccos(0.618) \approx 0.904) rad ((51.8^\circ)), and (x = 2\pi - 0.904 \approx 5.379) rad.
General:
[
x \approx 0.904 + 2k\pi, \quad x \approx 5.379 + 2k\pi
] Step 5 : (\cos x = 0
| Degrees | Radians | sin | cos | tan | |---------|---------|-----|-----|-----| | 0° | 0 | 0 | 1 | 0 | | 30° | π/6 | 1/2 | √3/2 | 1/√3 | | 45° | π/4 | √2/2 | √2/2 | 1 | | 60° | π/3 | √3/2 | 1/2 | √3 | | 90° | π/2 | 1 | 0 | ∞ |
Enunciado: Resuelve: (\sin^2 x + 3\cos x = 3) | Degrees | Radians | sin | cos
Solución:
Fixed: Muchos estudiantes olvidan que (\cos x = 2) no tiene solución real. Siempre verifica el rango.