ASTC rule in trigonometry (All Students Take Calculus)

Trigonometry ASTC

When you work with trigonometry, you’ll be dealing with four quadrants of a graph. The x and y axis divides up a coordinate plane into four separate sections.

ASTC is a memory-aid for memorizing whether a trigonometric ratio is positive or negative in each quadrant: [Add-Sugar-To-Coffee]

When you draw it out, it looks like this:

All, Sine, Tangent, Cosine chart.
ASTC chart

You can even use this diagram as a trigonometry cheat sheet. ASTC will help you remember how to reconstruct this diagram so you can use it when you’re met with trigonometry quadrants in your test questions.

In the above graphic, we have quadrant 1 2 3 4. In quadrant 1, both x and y are positive in value. In quadrant 2, x is negative while y is still positive. In quadrant 3, both x and y are negative. Lastly, in quadrant 4, x is positive while y is negative.

What this tells us is that if we have a triangle in quadrant one, sine, cosine and tangent will all be positive. In quadrant two, only sine will be positive while cosine and tangent will be negative. See how this is an easy way to allow you to remember which trigonometric ratios will be positive?

If you don’t like Add Sugar To Coffee, there’s other acronyms you can use such as:

All Students Take Calculus

All Stations To Central

Better yet, if you can come up with an acronym that works best for you, feel free to use it. As long as it contains ASTC in that order, you’ll remember the trig quadrants.

If you wanted to look further into trigonometric ratios, why not take a look and revise how the sine graph is graphed. You can also see how the cosine and tangent graphs look and what information you can get out of them.

What about the reciprocals of each trig function?

Recall that each of the three core trig functions have reciprocal identities. While these reciprocal identities are often used in solving and proving trig identities , it is important to see how they may fit in the grand scheme of the “All Students Take Calculus” rule

Most often than not, you will be provided with a “cheat sheet”, a sin cos tan chart outlining all the various trig identities associated with each of these core trigonometric functions. However, committing these reciprocal identities to memory should come naturally with the help of the memory aid discussed earlier above.

reciprocal identities
Reciprocal Identities

Looking at each reciprocal identity we can see that

cscθ\csc \theta is the reciprocal of sinθ\sin \theta