Taking the reciprocal of any value – whether it’s a number, letter, or fraction – means you flip the value over 1. Take for example the number 5. If we flip 5, it becomes a fraction 1/5. 1/5 is the “reciprocal” of 5. How about the fraction 7/8. If we flip 7/8, it becomes 8/7. 8/7 is the “reciprocal” of 7/8. Other examples are shown below:
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- Interestingly, as with all integers including the number 5 and –6 used above, they can be written as fractions. In other words, 5 is the same as 5/1, and –6 is the same as –6/1. This is why when you take their reciprocal, they become the fractions 1 over 5 and 1 over negative 6.
This same principal can be applied to the trigonometric functions learned previously. If we take the reciprocal of each trigonometric function – sin (θ), cos (θ), and tan (θ) – not only are they flipped, they also get a special name and abbreviation:
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The unfortunate part about this is that your calculator doesn’t have buttons designated for these reciprocal functions. So when asked to evaluate, let’s say sec (52.1°), you’ll have to remember that secant is 1 over cosine at angle 52.1°:
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In other words, you’ll type into your calculator 1 ÷ cos (52.1) to get 1.6279. See if you can evaluate these on your own:
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3.0422
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1.6426
Sometimes you will be given the ratio (typically as a decimal), and will be asked to find the angle that represents that decimal number. We already know how to do this using ordinary inverse trigonometric function, for example:
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But what about inverse reciprocal trigonometric functions?
To find theta (θ) when given cot θ = 1.7777 or csc θ = 4.2690, your calculator doesn’t have a button designated for the inverse of these reciprocal trigonometric functions either; for example, there’s no cot-1 or csc-1. Hence, you’ll need to remember that each of these equations are equivalent to their reciprocal versions:
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From the last step, you can easily use your calculator (tan-1).
Here’s a video demonstration of a few more examples. You’ll notice that in some examples, the prefix arc is placed in front of csc, sec, and cot. When you see this, it means csc-1, sec-1, and cot-1, respectively. Therefore, arccsc (4.2690) should be treated the same as:
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- In the unit to come, you’ll learn that there’s more to each of these answers. In fact, there are two positive angles between 0° and 360° for every ratio.