INSTRUCTOR: Welcome to a video that introduces the trigonometric functions in terms of right triangles. If you need to build a ramp that has an incline of 6 degrees and must reach a height of 2 feet, how long would the ramp have to be? Well, we can solve this type of problem using trigonometry. 

There are six trigonometric functions, and they are sine, cosine, tangent, cosecant, secant, and cotangent. Customarily, they are abbreviated using this notation in bold. Now, these trig functions can be defined in terms of particular ratios of sides of right triangles. Here are the sine, cosine, and tangent functions in terms of ratios of sides of right triangles. 

If we start at angle A, would be the ratio of the opposite side over the hypotenuse. So let's back up here for a second and identify. Remember that the hypotenuse is always opposite the right angle. So if you were to bisect the right angle, the side that you run into would always be the hypotenuse. 

Now, the opposite and adjacent sides will change based upon which angle you're referring to. So going back to angle A, if we bisect angle A, the side that we run into would be the opposite side. So in terms of angle A, this is the opposite side. So the sine of A is the ratio of the opposite over the hypotenuse side. So in this case, it would be A/C. 

Let's stick with angle A for a moment. The cosine of angle A would be the ratio of the adjacent side over the hypotenuse. Well, if this is the opposite side from angle A, this would be the adjacent side because this is always the hypotenuse. So the cosine of A would be the ratio of the adjacent side over the hypotenuse, or B/C. And tangent of angle A would be the ratio of the opposite side over the adjacent side. So here we'd have A/B/ 

OK, now if we look at the second row, notice they all refer to angle B. So that's going to change the orientation of the opposite and adjacent sides. So if we're referring to angle B, if we were to bisect this, now angle B would be the opposite side. And angle A would be the adjacent side. 

So notice how regardless of what angle we refer to, the ratio is always the same. Sine is the ratio of the opposite over the hypotenuse. Cosine is adjacent over hypotenuse, and tangent is always opposite over adjacent. 

So looking at angle B now, we have opposite over hypotenuse would be B/C for the sine B. Cosine B would be adjacent over hypotenuse, or A/C. And then tangent of B would be opposite over adjacent, or B/A/ One way to remember all of these would be to just remember this acronym SOH CAH TOA, which stands for the Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. 

Now, before, we learned that all right triangles with the same acute angles are similar. And so if we have that case, the trigonometric functions will produce the same ratios because we know that these triangles are proportional. So again, if you take a look at the large right triangle and the small green triangle, they share a common angle here. And they also have two right angles. 

So based upon angle-angle these are similar right triangles. So let's go ahead and label this full length. If that's 5 and that's 5, this would be 10. And if this is 4 and this is 4, this would be 8. So for the sine A, we would have the opposite length 6 over the hypotenuse length 10, 6/10. 

Now, if we look at the small triangle, opposite over hypotenuse would be 3/5, and we can see here that 6/10 is equivalent to 3/5, producing the same ratio. The same for the cosine of angle A, in the large triangle, we would have 8/10, and in the small triangle we would have 4/5, which, again, are equivalent ratios. And for the tangent of A, we would have the opposite over the adjacent, or 6 to 8 for the large right triangle. And the smaller would be 3/4, again equivalent. 

Now let's go ahead and define the three other trigonometric functions, and those are cosecant, secant, and cotangent. And what I want you to notice here is these are reciprocals of the sine, cosine, and tangent functions. So cosecant A would be the ratio of the hypotenuse to the opposite side. 

Remember that sine was opposite over hypotenuse. And the secant of A would be hypotenuse over adjacent, which is reciprocal of adjacent over hypotenuse, which is cosine. And the cotangent is adjacent over opposite, while tangent is opposite over adjacent. 

So again, all of these refer to angle A here. So this would be our hypotenuse. A would be the opposite, and B would be the adjacent. So we can see here, for cosecant A, we have the hypotenuse over the opposite, or C/A. Secant would be hypotenuse over adjacent, C/B. And cotangent would be adjacent over opposite, or B/A/ 

Let's determine the sign, cosine, and tangent of angle A. So here's our angle A. Let's go ahead and identify the three sides. Again, this is the hypotenuse. If this is angle A and we bisect it, this would be the opposite side. And this leaves this side as the adjacent side. 

So again, if we want to, we can use the acronym SOH CAH TOA if that's helpful. Now, before we find these three values, I notice that the length of the hypotenuse is missing. So we're going to have to apply the Pythagorean theorem to find this length. 

Remember, c squared equals a squared plus b squared, where c is the hypotenuse. So we would have c squared equals 5 squared plus 12 squared. This would be 25 plus 144. c squared equals 169. So c equals 13. You may recognize this as a 5, 12, 13 right triangle. 

So the sine of angle A would be the opposite over the hypotenuse, so 12/13. The cosine of angle A again is the adjacent over hypotenuse. So that would be 5/13. And tangent of A would be opposite over adjacent, or 12/5. And that's it. We found those trigonometric values. 

Let's try one more. Again, I can see, by the way, we're missing the length of this third side. So going back to our Pythagorean theorem, this is our hypotenuse. So we'd have 8 squared equals 4 squared plus x squared. 

So if we subtract 16 on both sides, we'd have 48. And square root both sides-- we only want the principal square root. So will we have x equals the square root of 48. But remember, 48 is 16 times 3. And the square root of 16 would be 4. So that's equal to 4 square root 3. 

OK, let's go ahead and set this up. Now, they are asking for cosecant, secant, and cotangent of angle B. So here's the angle we're referring to. So the 4 would be the opposite. And 4 square root 3 would be the adjacent side. 

Now, we can still use the acronym SOH CAH TOA to help us as long as we know which functions these are. The reciprocals of cosecant is the reciprocal of the sine function. So if sine is opposite over hypotenuse, cosecant would be hypotenuse over opposite. 

Well, starting at angle B, the hypotenuse is 8, and the opposite is 4. So we'd have 8/4, which equals 2. Secant is the reciprocal of cosine. So if cosine is adjacent over hypotenuse, we want the hypotenuse over the adjacent, which would be 8 over 4 square root 3. 

Now, this simplifies to 2 over square root 3. You may be required to rationalize this. This would rationalize to 2 square root 3 over 3. And lastly, cotangent is the reciprocal of tangent. So we would want the adjacent over the opposite, which would give us 4 square root 3 over 4, which just simplifies to square root 3. I hope you found this video helpful. Have a good day.