1

In a Cartesian coordinate system, there are 4 Quadrans. Lets take Quadrant 4 for reference. X there is positive and Y is negative. I understand that the radius in a unit circle is 1. This makes the sin of a 330 degree angle -1/2. But from the definition of sin = opposite/hypothenuse, it should always be positive since the length of a triangle should always be positive. I haven't seen a length of -2cm. So why is it different here? Why can't we just say that the sin of a +330 degree angle is 1/2. Why should we make it negative.

Robert Johnson
  • 31
  • 2
    Opposite of what over hypothenuse of what? So, what triangle has an angle of 330 degrees? – Jacky Chong Apr 16 '21 at 04:39
  • 2
    The definition via opposite/hypothenuse only works in the 1st quadrant, for other angles you have to tweak this definition. – lisyarus Apr 16 '21 at 04:39
  • This answer may (or may not) provide some insights. – Blue Apr 16 '21 at 04:42
  • https://en.wikipedia.org/wiki/Sine#Unit_circle_definition – Filthyscrub Apr 16 '21 at 04:43
  • 2
    Trigonometric functions are not defined via triangles; that identification came later, and it only works under suitable conditions; in particular, since you work on a right triangle with one angle of $90^{\circ}$, and the angles of a triangle add up to $180^{\circ}$, the correspondence can only work for angles $\alpha$ with $0\leq\alpha\leq 90^{\circ}$. To extend it to other angles, one uses the unit circle and the coordinates of the points on the circle, as this is related to the original connection of sines with half chords of the double angle. – Arturo Magidin Apr 16 '21 at 04:48
  • 1
    You should change your memorized definition to signed length of opposite/hypotenuse, etc. – Ted Shifrin Apr 16 '21 at 04:53
  • 1
    @TedShifrin: Ugh; if you’re going to do that, surely it’s better to just change it to “the coordinates of the point on the unit circle, $(\cos(\alpha),\sin(\alpha))$. – Arturo Magidin Apr 16 '21 at 04:56
  • I disagree with you, @Arturo. This is not the only place such notion show up. Very similar is the geometric interpretation of the dot product $x\cdot y = |x||y|\cos\theta$, where $|y|\cos\theta$ is the signed length of the projection of $y$ onto $x$. It's really the same issue. My pedagogical point stands, although perhaps not for a first-week trig student. – Ted Shifrin Apr 16 '21 at 05:03
  • "I haven't seen a length of -2cm." I haven't seen a an angle of 330 in a triangle, so all is good. – Carsten S Apr 16 '21 at 15:20

0 Answers0