alleryStart learning →Solving csc t = -2 on the interval from 0 to 4 pi, in which quadrants do all four solutions lie?
AThe first and second quadrants
BThe second and third quadrants
CThe third and fourth quadrants
DThe first and fourth quadrants
Answer & Solution
Correct answer: C. The third and fourth quadrants
1. Write csc t = -2 as 1 / sin t = -2.
2. Taking reciprocals gives sin t = -1/2.
3. A negative sine puts the angle in quadrant III or quadrant IV.
4. In the first turn the solutions are 7 pi/6 and 11 pi/6.
5. The interval runs to 4 pi, so the second turn adds 19 pi/6 and 23 pi/6.
6. All four solutions therefore lie in the third and fourth quadrants.
7. Quadrants I and II would require a positive sine, which contradicts the negative cosecant.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.5 Solving Trigonometric Equations_
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