Remember this picture? We used it to remember the signs of the three primary trigonometry ratios – Sine, Cosine and Tangent in the four quadrants. Each letter (of ASTC or CAST) represents the trigonometry function that is positive in each quadrant, e.g. Cosine is positive in the 4th quadrant etc., and All three are positive in the first quadrant.
Students sometimes used mnemonic devices to remember the order:
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but why do these basic trigonometric ratios take on different signs in different quadrants in the first place?
Sides of a right triangle
First we need to define these ratios (or trigonometry functions), and for that it would be easier if we name the sides of the triangle.
A triangle (with non-zero area) can have at most one right (90°) angle. For such right triangles, the three sides are given special names, with respect to one of non-90° angles (let’s call it angle a):
- Hypotenuse – the side across from the right angle
- Adjacent – the side adjacent to the angle a that is not the Hypotenuse
- Opposite – the side across from the angle a
For one specific angle a, e.g. a = 30° the three basic trigonometry functions – Sine, Cosine and Tangent, are ratios between the lengths of two of the three sides:
- Sine: sin(a) = Opposite / Hypotenuse
- Cosine: cos(a) = Adjacent / Hypotenuse
- Tangent: tan(a) = Opposite / Adjacent
That is all good when angle a is between 0° and 90°. What happens when the angle a is not acute?
Sine, Cosine and Tangent in Quadrant 1
Let’s place the right angle triangle on the standard cartesian axes for an angle a that is between 0° and 90° (we say that angle a is in Quadrant-1). In all of the figures below, the lengths are normalized to the hypothenuse, i.e. the hypothenuse is always of length 1.0, and is shown as the radius of the unit circle.
For an angle a that is in Quadrant 1, we see that the adjacent side lies on the “x-axis” and is in the direction of positive x. Meanwhile the opposite side is also in the direction of the positive y. Hence all three sides – adjacent, opposite and hypothenuse are all positive, Thus all (A) three ratios are positive.
Sine, Cosine and Tangent in Quadrant 2
When angle a is in Quadrant 2 (between 90° and 180°) however, the adjacent side is along the negative x- direction, while the opposite side is still in the positive y- direction. Hence, Cosine and Tangent are negative and only Sine (S) is positive.
Sine, Cosine and Tangent in Quadrant 3
When angle a is in Quadrant 3 (between 180° and 270°), both the adjacent and the opposite side are negative. Hence, Sine and Cosine are negative and since Tangent (T) is a division between two negative numbers, it is the only trigonometric function that is positive.
Sine, Cosine and Tangent in Quadrant 4
Finally, when angle a is in Quadrant 4 (between 270° and 360°), the adjacent side is back along the positive x- direction, while the opposite side is still in the positive y- direction. Hence, Sine and Tangent are negative and only Cosine (C) is positive.
Trigonometry Quadrant App
Here is a simple interactive app to illustrate the changes in signs of the three basic trigonometry ratios – Sine, Cosine Tangent in the four quadrants. Rotate the angle by mouse or touch to show angles in different quadrant and observe how the values of Sine, Cosine and Tangent changes with the sides of the right triangle.
This interactive manipulative is also a great way to show the three ratios as functions with respect to the angle a, i,e, by sweeping the angle from 0° and 360°, we see that the values of the three trigonometry functions describe the standard sine, cosine and tangent curves.