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Mathematics for Technology I (Math 1131)

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Durham College, Mathematics
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Reciprocal Trigonometric Functions

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:

x2reciprocal 1x26reciprocal  1 6
  • 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:

sinθreciprocal 1sinθ=cscθ   cosecantcosθreciprocal 1sinθ=secθ   secanttanθreciprocal 1tanθ=cotθ   cotangent

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°:

sec52.1°=1cos(52.1°)=1.6279

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:

cot 18.2°
Answer

3.0422

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csc 37.5°
Answer

1.6426

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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:

sin θ=0.4550sin1(0.4550)=θθ = 27.06°

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:

cot θ=1.77771tan θ=1.777711.7777=tan θ

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-1sec-1, and cot-1, respectively. Therefore, arccsc (4.2690) should be treated the same as:

csc θ=4.2690

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