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**?

**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**. 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**, and

^{-1}**cot**, respectively. Therefore,

^{-1}**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.