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How to Calculate an Angle Using Reciprocal Trigonometric Functions

Three trigonometric ratios — secant, cosecant, and cotangent — are called reciprocal functions because they're the reciprocals of sine, cosine, and tangent. These three functions open up three more ways in which you can solve equations in pre-calculus. The following list breaks down these functions and how you use them:

  • Cosecant, or

    image0.png

    is the reciprocal of sine. The reciprocal of a number a is 1/a, so

    image1.png

    Because

    image2.png

    you see that

    image3.png

    In other words, cosecant is the ratio of the hypotenuse to the opposite leg.

  • Secant, or

    image4.png

    is the reciprocal of cosine:

    image5.png

    Secant, in other words, is the ratio of the hypotenuse to the adjacent leg.

    A common mistake is to think that secant is the reciprocal of sine and that cosecant is the reciprocal of cosine, but the previous bullets illustrate the truth.

  • Cotangent, or

    image6.png

    is the reciprocal of tangent. (How's that for obvious?) You have the hang of this if you've looked at the previous bullets:

    image7.png

    Therefore,

    image8.png

    Remember that secant, cosecant, and cotangent are all reciprocals, but you typically won't find a button for them on your calculator. You must use their reciprocals — sine, cosine, and tangent. Don't get confused and use the sin–1, cos–1, and tan–1 buttons, either. Those buttons are for the inverse trig functions.

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