Given that $\cot(m)=0.75$ and $\cos(m)<0$, what is the value of $\sin(m)$?

Question

One of my homework problems is "Given that $\cot(m)=0.75$ and $\cos(m)<0$, what is the value of $\sin(m)$?" I keep getting $\sin m=\frac{-4}{5}=-.8$ which isn't an option

My options are:
A- $\ -.5625$
B-$\ -1.25$
C-$\ -.25$
D-$\ -.8$
E- None of the above

Edit: Because cotangent is positive and cosine is negative I know the angle is in the 3rd quadrant, I then used cotangent to get two of the dimensions, $3$ as the adjacent, and $4$ as the opposite. This gave me $5$ as the hypotenuse. Since sine is opposite divided by hypotenuse, I got $\sin(m)=\sin(4/5)=-.8$

Answer 1

By looking at the signs of $\cot(m)$ and $\cos(m)$ you can see from this link that $\sin(m)$ should be negative. It's great that you used 3 and 4 to get a 5 for the hypotenuse and get the 0.8 but I suggest you always beware of the differences between angle measures (or arc length measures) and line segment length measures. The trigonometric functions always use angles as arguments, so in $\sin(\theta)$, $\theta$ is an angle. When we are using formulas for calculating the value of a trigonometric function in an angle then we are using lengths of line segments. So, if we say $\sin(\theta) = {opposite \over hypotenuse}$ then $\theta$ is an angle but opposite and hypotenuse are length of line segments. Bottom line: lengths of line segments or rations of line segments (like ${opposite \over hypotenuse}$) should never be evaluated in a trigonometric function.

I suggest you check the graphs of trigonometric functions to get a feeling of how these functions behave with different values. I find this much more insightful than checking the tables of quadrants.

Answer 2

The cotangent is in the first or third quadrant but the cosine is less than zero so that limits it to the third quadrant where both the horizontal and vertical components are negative. It also means that $\cos\theta= -0.6$ because it's horizontal component is $-3$. Note, the hypotenuse is considered to be positive in all calculations. This means that the vertical component of the sine is $-4$ and that $\sin\theta=-0.8\space$ and you were correct in your calculation.