Reciprocal Trigonometric Functions on the Unit Circle - Online Tutor, Practice Problems & Exam Prep

In trigonometry, the reciprocal functions cosecant, secant, and cotangent are derived from sine, cosine, and tangent, respectively. Cosecant is defined as $1/y$, secant as $1/x$, and cotangent as $x/y$. Utilizing the unit circle, these values can be easily calculated, enhancing understanding of trigonometric relationships and their applications in various mathematical contexts.

Secant, Cosecant, & Cotangent on the Unit Circle

Hey, everyone. Up to this point, we've been working with 3 trig functions: sine, cosine, and tangent, on our unit circle. But remember, there are actually 3 other trig functions, our reciprocal trig functions: cosecant, secant, and cotangent. Now you may be worried that we're going to have to add a bunch of new information on our unit circle to find these reciprocal trig functions, but you don't have to worry about that at all because we can find the values of cosecant, secant, and cotangent using what's already on the unit circle. So let's go ahead and jump right in here.

Now these functions, remember, are called the reciprocal trig functions because they are simply one over our 3 original trig functions: sine, cosine, and tangent. So, for the cosecant of an angle, this is really just 1 over the sine of that angle. And on the unit circle, since we know that the sine of an angle is equal to that y value, the cosecant of the angle is just 1 over y. Then for the secant, since this is 1 over the cosine of that angle and we know that the cosine of any angle on our unit circle is just that x value, the secant of that angle is 1x. So looking at our unit circle, if I asked you to find the secant and cosecant of 60 degrees, you could simply take 1 over that x value and 1 over that y value, and you're done.

Now let's not forget the tangent here because we also have the cotangent as one of our reciprocal trig functions. And when we find the cotangent of an angle, this is just one over the tangent. Now, we know the tangent of an angle to be y over x on our unit circle, so we can simply flip that fraction to get the reciprocal, telling us that the cotangent of an angle is simply x over y. So again, if I asked you to find the cotangent of 60 degrees, you could go over to your unit circle, take your x value, divide it by your y value, and you'd be good to go.

Now with that in mind, let's go ahead and work through some examples together. So our first example here is the cosecant of π/6. Now we know that the cosecant is 1sinπ6, which we also know to be the y value. So going up to our unit circle here at π/6, looking at that y value of 1/2 here for the cosecant of π/6, I can take 1 over 1/2. Now whenever I take the reciprocal of a fraction, I'm really just flipping that fraction, so this gives me a value of 2. So the cosecant of π/6 is equal to 2.

Now let's look at the cotangent here. Now here we're asked to find the cotangent of π/4. Now we know that the cotangent of any angle is just our x value over our y value. So here we think about that again, x over y. Now we can go up to our unit circle at π/4, look at those x and y values, and simply divide them. So here we have the square root of 2 over 2 divided by the square root of 2 over 2. Now because these are the exact same value, we end up with an answer of 1. The cotangent of π/4 is just 1, which just so happens to be the exact same as the tangent of π/4 because those x and y values are the same.

Now let's look at one final example here. Here we have the secant of 0. Now the secant of 0, I know that this is just one over the x value at 0 degrees. So I can go up here to my unit circle at 0 degrees, identify that x value, and plug that in. So this is just going to be 1 over 1, which is of course just equal to 1. So the secant of 0 is just 1, and we're done here.

Now that we know how to find these reciprocal trig values from all of the info we already have on the unit circle, let's get some more practice. Thanks for watching, and let me know if you have questions.

Evaluate each expression.

$\cot\left(\frac{11\pi}{6}\right)$

• A. $\frac12$
• B. $-\frac{\sqrt3}{3}$
• C. $-\sqrt3$
• D. $2$

Evaluate each expression.

$\csc225\degree$

• A. $1$
• B. $-\frac{\sqrt2}{2}$
• C. $\sqrt2$
• D. $-\sqrt2$

Evaluate each expression.

$\sec\left(\frac{\pi}{3}\right)$

• A. $\frac12$
• B. $2$
• C. $\frac{\sqrt3}{2}$
• D. $\frac{2\sqrt3}{2}$

Here’s what students ask on this topic:

What are the reciprocal trigonometric functions and how are they related to sine, cosine, and tangent?

The reciprocal trigonometric functions are cosecant, secant, and cotangent. They are derived from the primary trigonometric functions sine, cosine, and tangent, respectively. Specifically, cosecant (csc) is the reciprocal of sine (sin), secant (sec) is the reciprocal of cosine (cos), and cotangent (cot) is the reciprocal of tangent (tan). Mathematically, this is expressed as:

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

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

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

How do you find the cosecant, secant, and cotangent of an angle using the unit circle?

To find the reciprocal trigonometric functions using the unit circle, follow these steps:

1. **Cosecant (csc)**: Identify the y-coordinate of the angle on the unit circle. Cosecant is the reciprocal of sine, so $\frac{1}{y}$.

2. **Secant (sec)**: Identify the x-coordinate of the angle on the unit circle. Secant is the reciprocal of cosine, so $\frac{1}{x}$.

3. **Cotangent (cot)**: Identify both the x and y coordinates of the angle. Cotangent is the reciprocal of tangent, so $\frac{x}{y}$.

What is the secant of 0 degrees and how is it calculated?

The secant of 0 degrees is calculated using the unit circle. At 0 degrees, the coordinates are (1, 0). Secant is the reciprocal of cosine, which is the x-coordinate. Therefore, secant of 0 degrees is:

$\frac{1}{1}=1$

So, sec(0°) = 1.

How do you find the cotangent of π/4 using the unit circle?

To find the cotangent of π/4 using the unit circle, identify the coordinates at π/4, which are $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Cotangent is the reciprocal of tangent, which is y/x. Therefore, cotangent of π/4 is:

$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$

So, cot(π/4) = 1.

What is the cosecant of π/6 and how is it calculated?

To find the cosecant of π/6 using the unit circle, identify the coordinates at π/6, which are $(\frac{\sqrt{3}}{2},\frac{1}{2})$. Cosecant is the reciprocal of sine, which is the y-coordinate. Therefore, cosecant of π/6 is:

$\frac{1}{\frac{1}{2}}=2$

So, csc(π/6) = 2.