Reciprocal Trigonometric Functions

Trigonometric functions are GREAT for solving triangles. That is, we can use sine, cosine and tangent, to find the missing parts of a triangle if we know just a handful of information about that triangle.  

The difficulties that arise are usually related to Algebra.  If you’re solving an equation that uses a Trigonometric function, say cosine, and you need to find the hypotenuse, it gets tricky.  This is where reciprocal functions come into play.  In practice and concept they’re not significantly different from the functions you already know, but the sometimes offer an easier path to finding unknown information.

Use the tabs below to read the notes, watch the video, download and play the PowerPoint.  Then, try the practice problems.

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At this point we have covered the basic Trigonometric functions.  The key idea is that the input is an angle, and the output is a ratio of sides.  If you need to find an angle, you use the inverse function. Then, the input is a ratio of sides, and the output is an angle.

We have also seen how right triangle trigonometry is used to find the sine, cosine, or tangent of angles obtuse, and even reflex, angles.  This is done through the use of a reference angle.  Visually, this is easy to understand by using the Unit Circle.

There are two key ideas that we have not yet discussed.  They do not restrict what it is you understand in anyway, but they’re related.  The two things we are going to discuss in this section are (1) reciprocal functions, and (2) radians.

Sine is wonderful, it’s our old favorite.  But, it is difficult to use sine to find the hypotenuse.  Same goes for cosine.

To solve this, you have to multiply both sides by C, and then divide both sides by the sign or thirty-two degrees.  Certainly it isn’t impossible, but if there’s a more efficient way, why not use it, right?

If you’re trying to solve for the hypotenuse, a reciprocal function is your friend!   Keep in mind, inverse functions and reciprocals are different. An inverse function “swaps” the input and the output.  A reciprocal is “flipped.”

These reciprocal functions were not introduced earlier because they are not conceptually different than the basic trigonometric functions.  Here’s where we get these reciprocal functions.

There’s not a need to do these steps every time we need to solve for the hypotenuse.  Instead, we can just define the reciprocal of sine, cosine and tangent as their own functions with reciprocal outputs.  That’s where reciprocal functions come into play and why they’re useful.  Let’s see an example.


While one single step might not seem like a big deal, this type of equation, where the unknown is in the denominator, is one of the “traps” in Algebra that students fall into.  The reason people get it wrong is because they confuse the order of division, which cannot be changed (there is no commutative property for division).

If we were going to solve the triangle for A, we could use tangent or cotangent.  Since we know the opposite side, and need to solve for the adjacent side, it would be easier to use cotangent because cotangent of an angle is the ratio of the adjacent : opposite.

Summary:  The reciprocal functions are just that, reciprocals of the trigonometric functions you already know and love.  They are useful when solving triangles that are missing the hypotenuse, and sometimes when you need to use tangent, but are missing the adjacent side. 

One confusing issue is their names.  Sine and Cosecant are reciprocals, while Cosine and Secant are reciprocals.  These names feel or sound misaligned.  Tangent and Cotangent are easy enough to keep straight!

Many calculators do not have reciprocal function operations programmed.  Do not confuse sin-1 with the reciprocal.  That is function notation.  The reciprocal would actually be (sin(θ))-1