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Memory Aid for Trigonometry Order Around the Quadrants

written by: Gillian Hendrie • edited by: Noreen Gunnell • updated: 1/6/2012

Are you never sure what order those trigonometric ratios come in? What's divided by what to produce what? Use these simple trigonometry mnemonics as memory aids to never have to worry about it again.

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    Mnemonics are a tried and proven method of remembering lists or the order of items that can be applied in many areas of learning. Here are a few mnemonics to help you with your trigonometry homework.

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    Order Around the Quadrants

    When the x,y-plane is divided into quadrants by the axes, each of the three trig ratios is positive in exactly two quadrants. All three ratios are positive in the first quadrant. In addition, in each of the remaining three quadrants, each ratio is positive exactly once more. But which is which? The correct order is:

    • 1st quadrant = ALL
    • 2nd quadrant = SINE only
    • 3rd quadrant = TANGENT only
    • 4th quadrant = COSINE only

    Now, how to remember this? There are at least two widely used methods, as detailed below.

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    First Method: CAST

    If you start in the fourth quadrant and go anticlockwise, the initial letters of the ratios and the word "all" spell CAST: (cosine, all, sine and tangent). Easy!

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    Second Method: ACTS

    This is the same idea as the first method, but here you start in the first quadrant and go clockwise: all, cosine, tangent, sine.

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    Third Method: Sentences

    If you don't like the idea of starting in the fourth quadrant - and it does seem a bit illogical to a mathematician! - or going clockwise (while the angle sizes increase in the opposite direction), then you can make up your own phrase or sentence. Beginning in the first quadrant and going anticlockwise, the initial letters would be A S T and C. Some sample sentences are:

    • All Students Take Classes.
    • Are Science Teachers Crazy?
    • Apple Skins Taste Crispy.

    You can see that you can easily make up your own fun memory sentence. Give it a go!

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    Download a summary revision sheet

    View the summary of the main points.