Hey everyone. So in the last video, we learned about the 3 main types of trigonometric functions, which are the sine, cosine, and tangent. And what we're going to be learning about in this video is 3 new trigonometric functions known as the cosecant, secant, and cotangent. Now just like the 3 trig functions we learned about in the previous video, these functions also have abbreviated versions of their names. We have the cosecant, secant, and cotangent. Now this might sound really scary and kinda complicated because there's just, like, a lot to remember and a lot of different things that we've learned about so far, but don't sweat it with these 3 trigonometric functions because what we're going to learn in this video is that these three functions are actually very closely related to the last 3 trigonometric functions we've learned about in a very simple way. So without further ado, let's get right into things to see what these new trig functions are all about.
Now the way that these functions are related to the previous 3 trig functions is they are reciprocals of the other trig functions. Now to be specific, we call these reciprocal identities. But all a reciprocal identity is is it's just a fancy way of saying that this thing is a flipped version of that thing. Now to understand this a little bit better, let's take a look at the cosecant for example. The cosecant is a reciprocal identity for the sine. So that means the cosecant is just a flipped version, meaning cosecant would be one over the sine. So if the sine of theta is opposite over hypotenuse, the cosecant of theta would be hypotenuse over opposite. So going to a right triangle, for example, we can look at our angle theta, and the hypotenuse or the long side would go on top of the fraction, and then the side opposite to the angle, which in this case is 3, would go on bottom of the fraction. So that's the idea of the cosecant. Now the secant is a reciprocal identity for the cosine. So that means that the secant of theta is 1cosθ. So if the cosine is adjacent over hypotenuse, then the secant is hypotenuse over adjacent. So going to our triangle, we have the hypotenuse or the long side, which is 5, and then this is divided by the side adjacent to our angle theta, which in this case is 4. So that's the secant of theta. Now it turns out that the tangent of theta also has a reciprocal identity, which is the cotangent of theta. So the cotangent of theta would be 1tanθ. And so if the tangent of theta is opposite over adjacent, then the cotangent of theta would be the side adjacent to our angle, which is 4 divided by the opposite side of the triangle, which is 3. Notice how the fractions for the cosecant, secant, and cotangent are literally just the flipped versions of the sine, cosine, and tangent. So this is the main idea and concept behind reciprocal identities.
Now something else I do want to mention is we talked about this identity in the previous video, how the tangent of theta is sine over cosine of theta. Well, this same rule of reciprocal identities applies to the cotangent of theta. So what we would do is flip this identity here, saying that cotangent is cosine over sine. So this is just another identity to watch out for when dealing with these types of trigonometric functions. Now to really make sure we have this concept solidified, let's actually try some examples where we have to use these new reciprocal identities. So in these problems, we're told to find the value of the trig function indicated given this right triangle. Now before I do anything, what I'm going to do is write down the memory tool we learned about in the previous video, which is SOHCAHTOA. SOHCAHTOA tells us how the 3 main trigonometric functions are related to the sides of a triangle. Now let's start by taking a look at example a. For example a, we are asked to find the secant of our angle x. Notice in this situation, we are with reference to angle x. Now when dealing with the secant, we learned that the secant is related to the cosine, and the cosine of x is going to be equal to adjacent divided by hypotenuse. Now if we go to our triangle and look at our angle x, the side adjacent to x is 5 and then this is divided by the hypotenuse or the longest side which is 13. So this is what the cosine is equal to, and since the secant is just the reciprocal of the cosine, we just flip this fraction. So it would become 13 over 5. So that is how you can find the secant of angle x using these reciprocal identities. But now let's take a look at example b where we are asked to find the cosecant of x. Well, we know that the cosecant is related to the sine. So we have the sine of x, and the sine of x is going to be equal to opposite over hypotenuse. Now we'll go into our angle x, and we can see the side opposite to x is 12, and then this is divided by the hypotenuse or the long side which is 13. So if we wanted to find the cosecant of x, which is just the flipped version of sine, we just flip this fraction and get 13 over 12. So that is the cosecant of our angle x. Now for example c, we're asked to find the cotangent of y. But notice this time we're looking at angle y, so we can no longer be with reference to x. So this is something we talked about in the previous video. You wanna make sure you are paying attention to which angle you are with reference to. So since we were with reference to angle y, we can find the cotangent of y by recognizing that the cotangent is the reciprocal for the tangent. So we have the tangent of y, we can see here is equal to opposite over adjacent. We'll go into the angle y, and here the side opposite the angle y is 5, then this is divided by the side adjacent to y, which is 12. And the flipped version of 5 over 12 would be 12 over 5, which is equal to the cosine of y. So the cosine of y is 12 over 5, and that is the answer to example c. So this is the main idea behind these 3 new trigonometric functions and how you can solve problems with them using these reciprocal identities. So hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.