Solve the problem.
-Which of the following trigonometric values are negative? I. $\sin \left( - 292 ^ { \circ } \right)$
II. $\tan \left( - 193 ^ { \circ } \right)$
III. $\cos \left( - 207 ^ { \circ } \right)$
IV. $\cot 222 ^ { \circ }$
A) II, III, and IV
B) I and III
C) III only
D) II and III

Question

Quizplus · Accepted Answer

To determine the sign of a trigonometric function, consider the function's angle and which quadrant it places the terminal side in when drawn in standard position. In standard position, angles are measured from the positive x-axis, with positive angles measured counterclockwise and negative angles measured clockwise.I. $\sin(-292^\circ)$ places the terminal side in the fourth quadrant, where sine is negative. However, since $-292^\circ$ is equivalent to $68^\circ$ (since adding $360^\circ$ gives a coterminal angle), and sine is positive in the first quadrant, this is incorrect in the explanation; sine is actually positive here.II. $\tan(-193^\circ)$ is equivalent to $\tan(167^\circ)$ (since adding $360^\circ$ gives a coterminal angle), placing the terminal side in the second quadrant, where tangent is negative.III. $\cos(-207^\circ)$ is equivalent to $\cos(153^\circ)$ (again, adding $360^\circ$ for a coterminal angle), placing the terminal side in the second quadrant, where cosine is negative.IV. $\cot(222^\circ)$ places the terminal side in the third quadrant, where cotangent is positive, because both sine and cosine are negative in the third quadrant, and cotangent is the reciprocal of tangent, which is positive when both sine and cosine are negative.Therefore, the trigonometric values that are negative are II. $\tan(-193^\circ)$ and III. $\cos(-207^\circ)$.

Solve the Problem sin⁡(−292∘)\sin \left( - 292 ^ { \circ } \right)sin(−292∘)

Solve the Problem $\sin \left( - 292 ^ { \circ } \right)$