Trigonometry · solution pairs
Where is the second solution?
On one turn of the circle, \(\cos\theta = c\) and \(\sin\theta = c\) each have two solutions when \(-1 < c < 1\). Cosine (余弦) and sine (正弦) pair them up in different ways.
1 · Predict
\(\cos\theta = -\tfrac12\) on \([0^\circ, 360^\circ)\). Where are the two angles?
Tap the circle twice, or type both angles in degrees. Then lock in your answer.
2 · Move the line
Drag the line and watch the pair
For \(\cos\theta = c\) the line is \(x = c\); for \(\sin\theta = c\) it is \(y = c\). Each solution is a point where the line meets the circle.
- θ₁
- θ₂
- Pair rule
- θ₁ + θ₂
- Mirror
On \([0^\circ, 360^\circ)\), a cos pair always adds to 360°. A sin pair adds to 180° only when \(c \ge 0\); try \(\sin\theta = -\tfrac12\). The pair rule is the part that never changes.
3 · Check
Three new ones
Type both solutions in degrees, in the form that sits inside the given interval.
\(\sin\theta = -\tfrac{\sqrt3}{2}\) on \([0^\circ, 360^\circ)\)
\(\cos\theta = \tfrac{\sqrt2}{2}\) on \((-180^\circ, 180^\circ]\)
\(\cos\theta = -\tfrac{\sqrt3}{2}\) on \([0^\circ, 360^\circ)\)
4 · Takeaway
\(\cos\theta = c\) pairs \(\theta\) with \(-\theta\), a mirror (对称) across the x-axis; \(\sin\theta = c\) pairs \(\theta\) with \(180^\circ - \theta\), a mirror across the y-axis. Then write each angle in the form that sits inside the interval (区间) you were given.