Final answer:
To answer the question, use the Pythagorean identity to find sin C, which is negative in the third quadrant. Then, use the identity for tan C which is the ratio of sin C to cos C. For angle C in quadrant III with cos C = -0.416, sin C = -0.909, and tan C = 2.188.
Step-by-step explanation:
The student question is asking to determine the values of sin C and tan C given that angle C is in quadrant III and cos C = -0.416. To find sin C, we use the Pythagorean identity sin² C + cos² C = 1. Since cos C is given, we can calculate sin C as follows:
sin² C = 1 - cos² C
sin² C = 1 - (-0.416)²
sin² C = 1 - 0.173056
sin² C = 0.826944
sin C = ±0.909
However, since the angle is in the third quadrant, where sine is negative, we take the negative value of sine.
sin C = -0.909
Now, to find tan C, we use the identity tan C = sin C/cos C.
tan C = -0.909 / -0.416
tan C = 2.188
Given that both sin C and tan C are negative in the third quadrant, the correct options should be sin C is negative and tan C is positive. Therefore, the answer is option a) sin C = -0.909, tan C = 2.188.