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

Understanding reciprocal trigonometric functions is essential for mastering trigonometry. Cosecant, secant, and cotangent are defined as the reciprocals of sine, cosine, and tangent, respectively. Specifically, cosecant is \( \frac{1}{\sin(\theta)} \), secant is \( \frac{1}{\cos(\theta)} \), and cotangent is \( \frac{1}{\tan(\theta)} \). Using the unit circle, these values can be easily derived from the coordinates. For example, cosecant of \( \frac{\pi}{6} \) yields 2, while cotangent of \( \frac{\pi}{4} \) equals 1. Mastery of these concepts enhances problem-solving skills in trigonometry.

Secant, Cosecant, & Cotangent on the Unit Circle

Hey, everyone. Up to this point, we've been working with 3 trigonometric functions: sine, cosine, and tangent on our unit circle. But remember, there are actually 3 other trigonometric 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 1sinθ 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 1y. Then for the secant, since this is 1cosθ 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 one over that x value and one 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 xy. 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 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 pi over 6. Now we know that the cosecant is 1sinπ6 which we also know to be the y value. So, at pi over 6 on our unit circle, looking at that y value of 1 half, for the cosecant of pi over 6 I can take 112. Now, flipping that fraction gives us a value of 2. So, the cosecant of pi over 6 is equal to 2.

Now let's look at the cotangent. Here we're asked to find the cotangent of pi over 4. Now we know that the cotangent of any angle is just our x value over our y value. So at pi over 4 on our unit circle, look at those x and y values, and simply divide them. Here we have 2222. Because these are the same value, we end up with an answer of 1. The cotangent of pi over 4 is just 1, which just so happens to be the exact same as the tangent of pi over 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 to my unit circle at 0 degrees, identify that x value, and plug that in. So this is just 11, 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 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 any 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 defined as the reciprocals of the primary trigonometric functions: sine, cosine, and tangent. 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:

$\csc \left(\theta \right)=\frac{1}{\sin }$

$\sec \left(\theta \right)=\frac{1}{\cos }$

$\cot \left(\theta \right)=\frac{1}{\tan }$

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, which is the sine value. The cosecant is the reciprocal of this value: $\csc \left(\theta \right)=\frac{1}{\sin }$.

2. **Secant (sec)**: Identify the x-coordinate of the angle on the unit circle, which is the cosine value. The secant is the reciprocal of this value: $\sec \left(\theta \right)=\frac{1}{\cos }$.

3. **Cotangent (cot)**: Identify the x and y coordinates of the angle on the unit circle. The cotangent is the ratio of the x-coordinate to the y-coordinate: $\cot \left(\theta \right)=\frac{x}{y}$.

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

The cosecant of π/6 is calculated by taking the reciprocal of the sine of π/6. On the unit circle, the sine of π/6 is the y-coordinate, which is 1/2. Therefore, the cosecant of π/6 is:

$\csc \left(\frac{\pi }{6}\right)=\frac{1}{\frac{1}{2}}$

Taking the reciprocal of 1/2 gives us 2. Thus, $\csc \left(\frac{\pi }{6}\right)=2$.

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 x and y coordinates at π/4. Both coordinates are $\frac{\sqrt{2}}{2}$. The cotangent is the ratio of the x-coordinate to the y-coordinate:

$\cot \left(\frac{\pi }{4}\right)=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$

Since the numerator and denominator are the same, the cotangent of π/4 is 1. Thus, $\cot \left(\frac{\pi }{4}\right)=1$.

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

The secant of 0 degrees is determined by taking the reciprocal of the cosine of 0 degrees. On the unit circle, the cosine of 0 degrees is the x-coordinate, which is 1. Therefore, the secant of 0 degrees is:

$\sec \left(0\right)=\frac{1}{1}$

Taking the reciprocal of 1 gives us 1. Thus, $\sec \left(0\right)=1$.