Question

use the controls of the figure to find the smallest value of t that satisfies all the following conditions: cos(t) = -\frac{\sqrt{3}}{2}, csc(t) = 2, and t < 0. use the interactive figure to find your answer. use the left and right arrow keys to move along a slider as needed. click here to launch the interactive figure. the smallest value of t is □ (type an exact answer, using π as needed.)

Explanation:

Step1: Analyze the value of \(\sin(t)\)

Since \(\csc(t)=\frac{1}{\sin(t)} = 2\), then \(\sin(t)=\frac{1}{2}\).

Step2: Determine the quadrant of \(t\)

We know that \(\cos(t)=-\frac{\sqrt{3}}{2}<0\) and \(\sin(t)=\frac{1}{2}>0\), so \(t\) is in the second quadrant. The general solution for \(\sin(t)=\frac{1}{2}\) is \(t = 2k\pi+\frac{\pi}{6}\) or \(t=(2k + 1)\pi-\frac{\pi}{6},k\in\mathbb{Z}\). For the second - quadrant solution, \(t=(2k + 1)\pi-\frac{\pi}{6}\).

Step3: Find the smallest negative \(t\)

Let \(k=-1\), then \(t=(- 2 + 1)\pi-\frac{\pi}{6}=-\pi-\frac{\pi}{6}=-\frac{7\pi}{6}\)

Answer:

\(-\frac{7\pi}{6}\)