I was studying about trigonometric functions and I found that while defining trigonometric functions of any angle ( +ve or -ve and of any size), they take up two mutually perpendicular lines and a revolving line and then construct a perpendicular on the horizontal line from any point on revolving line. Then they define the base, the perpendicular and the revolving line segment. Then trigonometric ratios of the angle theta made by the revolving line are described as -

$\sin \theta =$ Perpendicular/Revolving segment

$\cos \theta =$ Base /Revolving segment

$\tan \theta =$ Perpendicular/Base

And subsequently their respective cofunctions .

For understanding , what actually a sine of acute angle is , we can think it of as a ratio of the opposite side to hypotenuse of an angle contained in a right triangle. But the problem arose when I had to visualize sine of an angle greater than $90^\circ$. This is because I cannot relate it as a ratio of sides of a triangle which makes it hard to grasp practically. I know some of the applications of trigonometric functions of angle greater than $90^\circ$ like while calculating work done , so there must be an intuitive description of such trigonometric functions but I couldn't reveal any.

Please help me understand, what trigonometric ratios of such angles mean practically. I don't prefer any theoretical explanations like that of Cartesian plane or input circle approach. Thanks in advance for that!

  • $\begingroup$ If say $\frac{\pi}{2} < \theta < \pi$ you could consider the reference angle $0 < \pi - \theta < \frac{\pi}{2}$, noting that $\sin(\pi - \theta) = \sin \theta$ and $\cos(\pi - \theta) = - \cos \theta$ $\endgroup$ – joeb Mar 26 '17 at 15:48
  • $\begingroup$ I am not asking about the equality of trigonometric functions of two angles but asking for what does the sine of an angle greater than 90 degrees means. $\endgroup$ – Abhinav Dhawan Mar 26 '17 at 15:52
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    $\begingroup$ What you are talking about is a variation of unit (input?) circle definition of trigonometric functions. In the unit circle definition, the length of the "revolving segment" is $1$ unit and sine of the angle made by that segment is defined as the length of the perpendicular from the moving end of the segment to the horizontal. $\endgroup$ – Parth Mar 26 '17 at 19:02
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    $\begingroup$ You need to understand that the right triangle definition of trigonometric functions is valid only in the the range $(0,\frac{\pi}{2})$. You need to get your head around unit circle (if you want to relate it to the revolving segment definition) or other definitions of trig functions like the one involving series to get an intuitive idea of whats going around. $\endgroup$ – Parth Mar 27 '17 at 5:21
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    $\begingroup$ If I am not wrong, you are facing problem in visualising ratio of sides greater than $\frac{\pi}{2}$. As I said earlier, right triangle definition is only valid in range $(0,\frac{\pi}{2})$ so you are right that its difficult to visualise angles beyond this range. This is when you need to understand unit circle definition (if that is intuitive enough for you). If you use that definition, it will make sense what trig functions of such angles actually mean. You can't visualise them using only right triangle definition. Even the definition that you give in the question works perfectly. $\endgroup$ – Parth Mar 27 '17 at 6:07

The modern definition of the elementary trigonometric functions (in the form in which I learned it) is as follows. You have two perpendicular axes in a plane, viewed in an orientation such that one axis (the $x$ axis) runs horizontally from left to right and the other axis (the $y$ axis) is vertical. You construct a circle of radius $1$ with its center at the point of intersection of the two axes. This circle is called the unit circle. To define $\sin(\theta)$ and $\cos(\theta),$ travel a distance $\theta$ counterclockwise (that is, anticlockwise) around the unit circle, starting at the rightmost intersection of the unit circle and the $x$ axis; then $\sin(\theta)$ is the $y$-coordinate of the point you reach when you have traveled that distance and $\cos(\theta)$ is the $x$-coordinate of that point.

You can find a "rotating segment" in this construction by constructing a segment from the center of the unit circle to the point that you reach after traveling any distance. The business about dividing lengths by the length of the rotating segment is avoided by choosing units so that everything is measured relative to the length of the "rotating segment", including the distance traveled along the unit circle. As a result of this, you get sine and cosine functions for angles measured in radians.

So the sine and cosine functions are merely ways of converting a circular or turning motion into a linear (left-right or up-down) motion. For the first one-quarter of the first full circuit around the circle, we have the bonus feature that you can draw a right triangle with one leg along the $x$ axis, using the "rotating segment" as the hypotenuse, and now we have a bunch of facts about the relative lengths of sides of right triangles. But these facts about right triangles are just useful side effects of the definitions of the trigonometric functions, not the raison d'être of those functions.

For a real-life example of converting circular motion to linear motion in exactly this way, consider the Scotch yoke. This is a mechanism in which a crank pin (traveling in a circular path) is inserted into a rod through a crosswise slot in the rod so that as the crank turns, the crank pin pushes the rod back and forth. There are many illustrations of this mechanism that you can find; you can follow this link to a video, for example. There has even been an automotive engine built on this principle not very long ago.

  • $\begingroup$ Thanks... I liked the video and your approach towards thinking trigonometric functions as ways of converting circular motion to linear motion. $\endgroup$ – Abhinav Dhawan Nov 14 '17 at 13:41

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