Answer:
- sin(x) = -2(√6)/5
- csc(x) = -5/(2√6)
- tan(x) = 2√6
- cot(x) = 1/(2√6)
Step-by-step explanation:
In the third quadrant, sine and its inverse, cosecant, are negative. In that quadrant, tangent and its inverse, cotangent, are positive. The sine function always has a magnitude less than or equal to 1, while the cosecant function always has a magnitude at least 1.
The negative numbers on your list can be assigned to sin(x) and csc(x) based on their magnitudes. 2√6 = √24 < 5, so 2(√6)/5 < 1. The number with this magnitude is the sine; its inverse is the cosecant.
- sin(x) = -2(√6)/5
- csc(x) = -5/(2√6)
The tangent is the ratio of sine to cosine, so is ...
tan(x) = ((-2√6)/5)/(-1/5) = 2√6
and the cotangent is the inverse of that:
- tan(x) = 2√6
- cot(x) = 1/(2√6)
Of course, the secant is the inverse of the cosine, so would be -5. That is not one of the number choices, so sec(x) can be ignored.
