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Updated: Mar 28, 2021

One of many blunders that high school teachers commit (at-least in my case) is that the very first thing they would teach you about trigonometry, is the fact that trigonometric functions have certain sign conventions when we move from one quadrant to the other in a Cartesian co-ordinate system. In addition to that, they give you an "easy" way to memorize how this sign convention changes, i.e., "ALL-SILVER-TEA-CUPS".

  • ALL - Implying that all the functions are positive in the first quadrant.

  • SILVER - Implying that only the sine function is positive in the second quadrant.

  • TEA- Implying that only the tangent function is positive in the third quadrant.

  • CUPS- Implying that only the cosine function is positive in the fourth quadrant.

{NOTE: Bullet points are inclusive of reciprocals} And that is it, that's pretty much the stuff that is presented to you, which is quite a shame, because it definitely doesn't do a rightful justice to the intuition of how these functions have different signs in different quadrants. Intuition: Another standard introduction to trig is through triangles which persists in school and in reality, trigonometry is more about circles. Also, that intuition of trigonometry in terms of circles (more specifically unit circles, since sine and cosine cycle between 1 and -1) is more usefully while using stuff like Fourier transforms and problem solving in physics, problems involving harmonic motions, problems from electrodynamics and so on.

Imagine moving around along the circumference of a unit circle,

  1. Sin(x), 'x' is the input in this case and it is the measure of how far you have moved along the circle. As you can see from the desmos graph present above, as we move from the right most side of the circle which has a unit radius, and your journey around the circle is at a constant rate. As function of how far you have moved, sine graphs out the height at a certain input along the y-axis. The y-axis as depicted in the graph, is the distance between you or a point on the unit circle and the x-axis. As a result, as you transverse around the circumference the wave starts to oscillate with a constant period. From the picture it is quite clear the reason why a sine graph starts from zero, it is because by definition sine is giving you the height (y co-ordinates) and when it starts off from the right side, it starts with zero, since, the height or the output is zero and it gradually increases.

2.Cos(x), the cosine is defined very similarly, but this time it gives you the x co-ordinate values or the distance to the vertical line as you moved around that unit circle. It starts off at 1 and as you move around the circumference, recording the distance from the horizontal line, which gets lower, reaching to the value of "-1" before it starts increasing again.

As a consequence of the two illustrations above, we can say that sine and cosine can be thought of as distances from the two co-ordinates, resulting from the various inputs, i.e., 'x'.

Let's take an example, for the input values, look at the cosine graph. At the input pi, we have traveled halfway around the circle to the value "-1", resulting in the negative value of that function in the second quadrant, for any input the output of cosine after the first quadrant lies on the negative x-axis. Similarly, look at the sine graph for the input Pi, the output is zero and for any other input value lesser than pi, the vertical distance or the distance to the y-axis co-ordinate is always going to be positive between inputs of zero and pi.

One more thing, the point where the two distances meet, if that point is joined to the origin a right triangle in formed and so some the connection between sine and cosine functions, with triangles.

The intuition behind the sign conventions can enlighten us, when we think of sine and cosine as functions that help us to find the position of points along a unit circle, which in many people's sense is more important than merely knowing how to compute these values. since, we have computers to do that for us.

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