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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.
काम में एक अच्छी मिसाल: sin(x) = 1/2
चरण द्वारा कदम
- \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.
जवाब दिखाएँ
x = \frac{\pi}{6} ,\; x = \frac{5 \pi}{6}