Solve sin(3x) = 1
Trigonometric equation, worked out line by line the way a teacher would write it.
Answer
| General solution | x = π/6 + 2π/3·n (n any integer) |
Step-by-step solution
6 steps-
1 Given\sin{\left(3 x \right)} = 1
Solve for x.
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2 A special value: read it off the unit circle
On the unit circle the point at angle θ has coordinates (cos θ, sin θ). The values 0, 1 and −1 sit at the quarter turns, so those angles can be read off directly.
3 x = \frac{\pi}{2} -
3 Solve each family for x
The angle inside the function is not x itself, so each family of angles is solved for x like a linear equation; dividing by the coefficient of x also divides the period.
x = \frac{\pi}{6} + \frac{2 \pi n}{3} -
4 General solution (n is any integer)
Sine and cosine repeat every full turn (360° or 2π) and tangent every half turn, so adding any whole number n of periods gives another solution. n can be any integer, positive, negative or 0.
x = \frac{\pi}{6} + \frac{2 \pi n}{3},\quad n \in \mathbb{Z} -
5 The solutions in one turn, 0 ≤ x < 2πx = \frac{\pi}{6},\; \frac{5 \pi}{6},\; \frac{3 \pi}{2}
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6 Check\begin{aligned}x = \frac{\pi}{6}:\quad 1 = 1\quad\checkmark\\ x = \frac{5 \pi}{6}:\quad 1 = 1\quad\checkmark\\ x = \frac{3 \pi}{2}:\quad 1 = 1\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| pi/6 | 1 | 1 | ✓ |
| 5pi/6 | 1 | 1 | ✓ |
| 3pi/2 | 1 | 1 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.