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 $\sin(m)=\sin(4/5)=-.8$

  • $\begingroup$ Try drawing a unit circle, and explicitly draw the angle out, double check the definitions of trig functions. Please explain why you get $\sin (-4/5)$, as I cannot imagine how. $\endgroup$ – Trebor Aug 19 '18 at 7:20
  • $\begingroup$ I think OP means $\sin m= -0.8$ $\endgroup$ – Mohammad Zuhair Khan Aug 19 '18 at 7:26
  • 1
    $\begingroup$ isn't $-\frac{4}{5} = -\frac{8}{10} = -0.8$? $\endgroup$ – Ronald Aug 19 '18 at 7:32
  • 2
    $\begingroup$ You should have written $\sin m = -\frac{4}{5} = -0.8$. $\endgroup$ – N. F. Taussig Aug 19 '18 at 7:35
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    $\begingroup$ You should also change the final line. However, the key point is that your revised answer is option D. $\endgroup$ – N. F. Taussig Aug 19 '18 at 7:40

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.


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