What Quadrant Is Tan Negative

Remember this picture? Nosotros used it to think the signs of the 3 primary trigonometry ratios – Sine, Cosine and Tangent in the four quadrants. Each letter of the alphabet (of ASTC or CAST) represents the trigonometry function that is positive in each quadrant, due east.k. Cosine is positive in the quaternary quadrant etc., and All three are positive in the first quadrant.

Students sometimes used mnemonic devices to remember the social club:

All Southcience Teachers Care
All Students Take Chemistry
etc

just why do these basic trigonometric ratios take on different signs in different quadrants in the first place?

Sides of a right triangle

First nosotros demand 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-zip expanse) tin can have at most 1 right (90°) angle. For such right triangles, the iii sides are given special names, with respect to one of not-ninety° angles (allow'due south call information technology angle a):

Hypotenuse – the side across from the right angle
Adjacent – the side adjacent to the angle a that is non the Hypotenuse
Reverse – the side across from the bending a

For i specific angle a, due east.thou. a = xxx° the 3 basic trigonometry functions – Sine, Cosine and Tangent, are ratios betwixt the lengths of ii of the three sides:

Sine: sin(a) = Contrary / Hypotenuse
Cosine: cos(a) = Adjacent / Hypotenuse
Tangent: tan(a) = Opposite / Adjacent

That is all good when bending a is betwixt 0° and 90°. What happens when the angle a is non acute?

Sine, Cosine and Tangent in Quadrant one

Let'southward place the correct angle triangle on the standard cartesian axes for an angle a that is betwixt 0° and 90° (we say that angle a is in Quadrant-ane). In all of the figures below, the lengths are normalized to the hypothenuse, i.eastward. the hypothenuse is always of length 1.0, and is shown every bit the radius of the unit circle.

For an angle a that is in Quadrant i, we see that the adjacent side lies on the "x-axis" and is in the management of positive ten. Meanwhile the opposite side is likewise in the management of the positive y. Hence all three sides – adjacent, reverse and hypothenuse are all positive, Thus all (A) three ratios are positive.

Sine, Cosine and Tangent in Quadrant 2

trigonometry functions sine, cosine and tangent in the 2nd quadrant

When angle a is in Quadrant 2 (between xc° and 180°) however, the adjacent side is along the negative x- direction, while the contrary side is still in the positive y- management. Hence, Cosine and Tangent are negative and only Sine (S) is positive.

Sine, Cosine and Tangent in Quadrant 3

trigonometry functions sine, cosine and tangent in the 3rd quadrant

When angle a is in Quadrant 3 (between 180° and 270°), both the side by side and the opposite side are negative. Hence, Sine and Cosine are negative and since Tangent (T) is a division between ii negative numbers, it is the only trigonometric function that is positive.

Sine, Cosine and Tangent in Quadrant 4

trigonometry functions sine, cosine and tangent in the 4th quadrant

Finally, when bending a is in Quadrant four (between 270° and 360°), the adjacent side is dorsum along the positive x- direction, while the reverse side is still in the positive y- management. Hence, Sine and Tangent are negative and merely Cosine (C) is positive.

Trigonometry Quadrant App

Here is a uncomplicated interactive app to illustrate the changes in signs of the three basic trigonometry ratios – Sine, Cosine Tangent in the iv quadrants. Rotate the bending by mouse or bear on to show angles in unlike quadrant and observe how the values of Sine, Cosine and Tangent changes with the sides of the right triangle.

This interactive manipulative is besides a slap-up way to evidence the three ratios as functions with respect to the angle a, i,e, by sweeping the angle from 0° and 360°, nosotros see that the values of the three trigonometry functions describe the standard sine, cosine and tangent curves.