Solve sin(x) = −1
Trigonometric equation, worked out line by line the way a teacher would write it.
Answer
| General solution | x = 3π/2 + 2π·n (n any integer) |
Step-by-step solution
5 steps-
1 Given\sin{\left(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.
x = \frac{3 \pi}{2} -
3 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{3 \pi}{2} + 2 \pi n,\quad n \in \mathbb{Z} -
4 The solutions in one turn, 0 ≤ x < 2πx = \frac{3 \pi}{2}
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5 Check\begin{aligned}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 | |
|---|---|---|---|
| 3pi/2 | −1 | −1 | ✓ |
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