Evaluate the expressions using reference angles. $\csc \left( - 840 ^ { \circ } \right)$ $\cot \left( - 840 ^ { \circ } \right)$
A) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 2 \sqrt { 2 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 2 } } { 2 }$
B) $\csc \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$
C) $\csc \left( - 840 ^ { \circ } \right) = - \frac { \sqrt { 2 } } { 2 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 2 } } { 3 }$
D) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 1 } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { 1 } { 3 }$
E) $\csc \left( - 840 ^ { \circ } \right) = - \frac { 2 \sqrt { 3 } } { 3 }$ $\cot \left( - 840 ^ { \circ } \right) = \frac { \sqrt { 3 } } { 3 }$

Question

Quizplus · Accepted Answer

First, simplify the angle by finding its equivalent in the first rotation (0 to 360 degrees). -840 degrees + 360 degrees (x2) = -120 degrees, which is equivalent to 240 degrees in standard position. The reference angle for 240 degrees is 60 degrees, which is in the third quadrant where both csc and cot are negative. Therefore, $\csc(240^\circ) = -\frac{2\sqrt{3}}{3}$ and $\cot(240^\circ) = \frac{\sqrt{3}}{3}$, matching the values in choice E.

Evaluate the Expressions Using Reference Angles csc⁡(−840∘)\csc \left( - 840 ^ { \circ } \right)csc(−840∘)

Evaluate the Expressions Using Reference Angles $\csc \left( - 840 ^ { \circ } \right)$