True or false: csc(7π/6) = −2.

Step-by-step solution

1. 1

   Recall the definition of cosecant. Cosecant is not an inverse function but a reciprocal:

   $\csc\theta = \frac{1}{\sin\theta}$

   So the whole question reduces to finding $\sin\left(\tfrac{7\pi}{6}\right)$ exactly.

2. 2

   Locate the angle and find its reference angle. Write the angle as a rotation past $\pi$:

   $\frac{7\pi}{6} = \pi + \frac{\pi}{6}$

   An angle just past $\pi$ (that is, past $180^{\circ}$) sits in quadrant III, and its reference angle is $\tfrac{\pi}{6} = 30^{\circ}$.

3. 3

   Apply the quadrant sign. In quadrant III both $x$ and $y$ coordinates on the unit circle are negative, so sine is negative there. With $\sin\tfrac{\pi}{6} = \tfrac12$,

   $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$

   The unit-circle point is $\left(-\tfrac{\sqrt3}{2},\, -\tfrac12\right)$, and the $y$-coordinate is the sine.

4. 4

   Take the reciprocal.

   $\csc\left(\frac{7\pi}{6}\right) = \frac{1}{-\tfrac12} = -2$

   A reciprocal keeps the sign, so a negative sine gives a negative cosecant — and since $\left|\sin\right| = \tfrac12 < 1$, the cosecant must have magnitude greater than $1$, consistent with $2$.

5. 5

   Answer the true/false question. The statement matches the computed value exactly, so it is true. A numerical check gives $1/\sin(7\pi/6) = -2.000000000000001$, the tiny residue being floating-point error rather than a genuine discrepancy.