Identidades reciprocas

By this point you're probably familiar with how sine, cosine, and tangent are defined. These are the "foundational trig functions" because we define the other trig functions using these ones, and the most commonly used trigonometric identities involve sine, cosine, and tangent.

From here on we will abbreviate sine as $\sin,$ cosine as $\cos,$ and tangent as $\tan.$

Now let's introduce the reciprocal functions. The secant is the reciprocal of cosine, and is abbreviated $\sec.$ We define $\sec(x)$ as

$$\sec(x) = \frac{1}{\cos(x)}.$$

The cosecant is the reciprocal of sine, and is abbreviated $\csc.$ We define $\csc(x)$ as

$$\csc(x) = \frac{1}{\sin(x)}.$$

Finally, the cotangent is the reciprocal of tangent, and is abbreviated $\cot.$ We define $\cot(x)$ as

$$\cot(x) = \frac{1}{\tan(x)}.$$

In practice, we often use this formula combined with the fact that $\tan(x) = \frac{\sin(x)}{\cos(x)}.$ Thus another very useful identity to know is

$$\cot(x) = \frac{1}{\frac{\sin(x)}{\cos(x)}} = \frac{\cos(x)}{\sin(x)}.$$

One other thing to keep in mind: all three definitions, those of secant, cosecant, and cotangent, involve dividing something that might be zero. For instance, if $\sin(x) = 0$ then $\csc(x)$ isn't defined because we'd be dividing by zero. This causes secant, cosecant, and cotangent to have "holes" in their domains. To see what I mean, take a look at graphs of all three functions.

(Video) What are the reciprocal identities of trigonometric functions

by Brian McLogan

Watch this video by Brian Logan about the reciprocal identities in trigonometry!

To summarize, the reciprocal identities are:

$$\sin{\theta} = \frac{1}{\csc{\theta}}$$ $$\csc{\theta} = \frac{1}{sin{\theta}}$$

and

$$\cos{\theta} = \frac{1}{\sec{\theta}}$$ $$\sec{\theta} = \frac{1}{\cos{\theta}}$$

and $$\tan{\theta} = \frac{1}{\cot{\theta}}$$ $$\cot{\theta} = \frac{1}{\tan{\theta}}$$

Trigonometry: Reciprocal Identities

We've already covered the tangent identity! Now we will move on in our series of definition identities.

We've written out these reciprocal identities as quotient identities! That means that we have these formulas dictated with "sine", "cosine", and "tangent", etc.

In trig, reciprocal functions are pairs of functions that "flip" each other. That means if you know one, you know the other because you can take $1$ divided by the function.

This concept is useful when solving equations and understanding the relationships between trig functions.

####Looking Further

In the next couple of lessons, you will learn about the pythagorean identity and how to derive identities.