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Trigonometric equations
Reference angles and the infinitely many solutions.
sin x = ½ is true at π/6, at 5π/6, and again every full turn. Solve on one period from the unit circle, then add 2πn for every integer n. The graph shows why the solutions repeat.
வேலை செய்த உதாரணம்: sin(x) = 1/2
படிப்படியாக
- \sin{\left(x \right)} = \frac{1}{2}
Start from the equation as given.
- \sin{\left(x \right)} - \frac{1}{2} = 0
Bring everything to one side.
- \sin{\left(x \right)} - \frac{1}{2} = 0
Isolate the trig function, then read the reference angle off the unit circle. Solutions repeat every period.
- \left\{2 n \pi + \frac{\pi}{6}\; \middle|\; n \in \mathbb{Z}\right\} \cup \left\{2 n \pi + \frac{5 \pi}{6}\; \middle|\; n \in \mathbb{Z}\right\}
General solution over the reals (n is any integer).
- x = \frac{\pi}{6} \approx 0.52360 ,\; x = \frac{5 \pi}{6} \approx 2.6180
Solve for the variable.
விடை தெரியப்படுத்து
x = \frac{\pi}{6} ,\; x = \frac{5 \pi}{6}