Have you ever heard of the expression SOH CAH TOA? What do you think this expression means? In this lesson, we're going to focus on right triangle trigonometry. Let's say if this is the angle theta. Now, there's three sides of this triangle that you need to be familiar with.
Opposite to theta, this is the opposite side. And next to the angle theta is the adjacent side. and across the box, or the right angle of the triangle, which is the hypotenuse. That's the longer side of the triangle. Now, if you recall, this is A, B, and C.
The Pythagorean theorem applies to right triangles. A squared plus B squared is equal to C squared. We're not going to focus on that too much, but just be familiar with that equation.
Now let's talk about the six trig functions in terms of sine, cosine, tangent, opposite, adjacent, hypotenuse. Sine theta, according to Sokotoa, s is for sine, o is for opposite, h is for hypotenuse. sine theta is equal to the opposite side divided by the hypotenuse.
Cosine theta is equal to the adjacent side divided by the hypotenuse. Ka is for cosine is adjacent over hypotenuse. And tangent theta TOA is equal to the opposite side divided by the adjacent side.
So that's the tangent ratio. It's opposite over adjacent. Now we know that cosecant is 1 over sine.
So cosecant is basically hyper... hypotenuse divided by the opposite side. You just need to flip this particular fraction. Secant is the reciprocal of cosine, so secant is going to be hypotenuse divided by the adjacent side. Cotangent is the reciprocal of tangent, so if tangent is opposite over adjacent, cotangent is adjacent divided by the opposite side.
Now let's say if we're given a right triangle and we have the value of two sides. Let's say this is 3 and this is 4. And here's the angle theta. Find the missing side of this right triangle. And then find the values of all six trigonometric functions. Sine, cosine, tangent, secant, cosecant, cotangent.
Now to find the missing side, we need to use the Pythagorean theorem. a squared plus b squared is equal to c squared. So a is 3, b is 4. And we've got to find the missing side, c, which is dipotens. 3 squared is 9, 4 squared is 16. 9 plus 16 is 25. And if you take the square root of both sides, you can see that the hypotenuse is 5. Now it turns out that there are some special numbers.
There's the 3, 4, 5 right triangle, the 5, 12, 13 right triangle, the 8, 15, 17 right triangle, and the 7, 24, 25 right triangle. And any whole number ratios or multiples of these numbers will also work. For example, if we multiply this by 2, we'll get 6, 8, 10. That can also work. Or, if you multiply it by 3...
you get the 9-12-15 triangle. If you multiply this one by 2, you get the 10-24-26 triangle. Those are also special triplets.
They work with any right triangle. Now, some other numbers that are less common, but you might see, are the 9-40-41 triangle and the 11-60-61. So if you see some of these numbers, you can find their missing side quickly if you know them. So now let's finish this problem. So what is the value of sine theta?
So according to Sokotoa, we know that sine theta is equal to the opposite side divided by the adjacent side. In the part SOH. Opposite to theta is 4. and the hypotenuse is 5. So therefore, sine theta is going to be 4 divided by 5. Now, cosine theta is equal to the adjacent side divided by the hypotenuse. So we said 4 is the opposite side, 5 is the hypotenuse, and 3 is the adjacent side. So in this case, it's going to be 3 divided by 5. So that's the value of cosine.
Now let's find the value of tangent. Tangent theta, according to Tola, is equal to the opposite side divided by the adjacent side. So that's going to be 4 divided by 3. So that's the value of tangent.
Now once we have these three, we can easily find the other three. To find... Cosecant, it's 1 over sine, so just flip this fraction.
It's going to be 5 over 4. And secant is the reciprocal of cosine, so flip this fraction. Secant's going to be 5 over 3. Cotangent is the reciprocal of tangent. So if tangent's 4 over 3, cotangent is going to be 3 over 4. And that's how you can find the value of the six trigonometric functions.
Let's try another problem. So let's say this is theta again, and this side is 8, and this side is 17. Find the missing side, and then use the completed triangle to find the value of the 6 trigonometric functions. So go ahead and pause the video and work on this problem.
So first, we need to know that this is the 8, 15, 17 triangle. If you ever forget, you can fall back to this equation. So, A is 8. We're looking for the missing side B, and the hypotenuse is 17. 8 squared is 64, and 17 squared is 289. 289 minus 64 is 225. And we need to take the square root of both sides.
And the square root of 225 is 15, which gives us the missing side of the triangle. So now, go ahead and find the value of sine theta, cosine theta, tangent theta, and then cosecant theta, secant theta, and cotangent theta. So using SOHCAHTOA, we know that sine is equal to the opposite side divided by the hypotenuse. So let's label all the three sides.
17 is the hypotenuse, 8 is the adjacent side, and opposite to theta is 15. So opposite over hypotenuse, this is going to be 15 divided by 17. So that is the value of sine theta. Now, cosine theta is going to be equal to the adjacent side divided by the hypotenuse. So the adjacent side is 8, the hypotenuse is 17. So, cosine theta is 8 over 17. Tangent, based on TOA, is going to be opposite over adjacent.
So, opposite is 15, adjacent is 8. Therefore, tangent is going to be 15 divided by 8. Now, cosecant is the reciprocal of sine. So, if sine theta is 15 over 17, cosecant is going to be 17 over 15. Secant is the reciprocal of cosine. So, if... If cosine is 8 over 17, secant is 17 over 8. You just got to flip it. And cotan is a reciprocal of tangent.
So cotan is going to be 8 over 15. Just flip this fraction. And now we have the values of the six trigonometric functions. And that's all you got to do.
So here's a different problem. So let's say here's our right angle, and this time this is theta. And let's say the hypotenuse is 25, and this side is 15. Find the missing side, and then go ahead and find the value of the six trigonometric functions.
So this is going to be similar to the 3, 4, 5 triangle. Notice that if we multiply everything by 5, we'll get 2 of the 3 numbers that we need. 3 times 5 is 15, 4 times 5 is 20, 5 times 5 is 25. So we have the 15, and we have the 25. Therefore, the missing side must be 20. And you can use the Pythagorean theorem to confirm this if you want to. So now, let's go ahead and find the value of sine theta. So opposite to theta is 20. The hypotenuse is always across the box, it's the longer side.
So 27 is the hypotenuse, and adjacent to 15, or right next to it, is 15. I mean, adjacent to theta is 15. Now sine theta, we know it's opposite divided by hypotenuse, so it's 20 over 25. which reduces to 4 over 5, if we divide both numbers by 5. 20 divided by 5 is 4, 25 divided by 5 is 5. Cosine theta is adjacent over hypotenuse, so that's 15 divided by 25, which reduces to 3 divided by 5. Tangent theta is opposite over adjacent, so 20 over 15, which becomes... If you divide by 5, that's going to be 4 over 3. Now, cosecant is the reciprocal of sine, so it's going to be 5 over 4, based on this value. And if cosine is 3 over 5, then secant, the reciprocal of cosine, has to be 5 divided by 3. Now, if tangent is 4 over 3, cotangent has to be 3 divided by 4. And so that's it for this problem.
Consider the right triangle. In this right triangle, find the missing side. In this case, find the value of x.
Let's say the angle is 38 degrees and this side is 42. So what trig function should you use in order to find the value of x? Should we use sine, cosine, or tangent? Well, relative to 38. We have the opposite side, which is x, and the adjacent side, which is 42. So tangent, we know it's opposite over adjacent. So therefore, tangent of the angle 38 degrees is equal to the opposite side x divided by the adjacent side 42. So in order to get x by itself, we need to multiply both sides by 42. So these will cancel.
So therefore, x... is equal to 42 tangent of 38. So we need to use the calculator to get this answer. And make sure your calculator is in degree mode.
So tan 38, which is 0.7813, and let's multiply that by 42. So this will give you an x value of 32.8. Now let's try another example. Feel free to pause the video to work on each of these problems, by the way. So let's say this angle is 54 degrees, and we're looking for the value of x, and the hypotenuse is 26. Which trig function should we use? Sine, cosine, or tangent?
So opposite to the right angle, we know it's the hypotenuse. And x is on the adjacent side relative to 54. So cosine is associated with adjacent and hypotenuse. So therefore, cosine of the angle 54 is equal to the adjacent side x divided by the hypotenuse of 26. So to get x by itself, we've got to multiply both sides by 26. So, therefore, X is equal to 26 cosine of 54 degrees.
Cosine 54 is 0.587785. If we multiply that by 26, this will give us the value of X, which is 15.28. Here's another one that we could try. Let's say the angle is 32 degrees and the hypotenuse is x and this is 12. So notice that 12 is opposite to 32 and we have the hypotenuse. So this time we need to use the sine function.
Sine of the angle 32 is equal to the opposite side 12 divided by x. So in this case, what can we do to find the value of x? What would you do? What I would do is cross multiply. So 1 times 12 is 12. And this is going to equal x times sine 32. Next, I recommend dividing both sides by sine 32. Sine 32 divided by itself is 1. So therefore, x is equal to 12 over sine 32. 12 divided by sine 32 is 22.64.
So that's the value of x in this particular problem. Now let's work on another problem. So this time, we need to find the angle theta.
And we're given these two sides. So 5 is opposite to the angle. and 4 is adjacent to it.
So what trig function can relate theta, 4, and 5? We know tangent is opposite over adjacent. So tangent of the angle theta is equal to the opposite side.
side, which is 5, divided by the adjacent side, 4. So how can we find the value of the angle theta? If tangent theta is 5 over 4, then theta is going to be the inverse tangent or arc tangent. of 5 over 4. And you simply have to type this in your calculator.
So type in R tan 5 over 4. And you should get an angle of 51.34 degrees. So that's how you can find the missing angle. Let's try another example.
Feel free to pause the video and find a missing angle. So in this case, we have the adjacent side, and we have the hypotenuse. So therefore, this is associated with cosine.
Cosine theta is equal to the adjacent side divided by the hypotenuse. So if cosine theta is equal to 3 divided by 7, theta is going to be arc cosine 3 over 7. And once again, you have to use a calculator to figure this out. Because without a calculator, how do we know what this answer is?
And this is going to be 64.62 degrees. So here's another one for you. Let's say this is 5 and this is 6. Go ahead and find the value of theta. So the hypotenuse is 6, opposite to theta is 5. So we know sine is associated with opposite and hypotenuse. Sine theta is equal to the opposite side, which is 5, divided by the hypotenuse, which is 6. Therefore, theta is the arcsine or inverse sine of 5 over 6. and so the angle is going to be 56.44 degrees and that's it that's all you gotta do to find the missing angle of a right triangle for those of you who might be interested in my trigonometry course here's how you can access it so go to udemy.com and once you're there Enter into the search box, Trigonometry.
Now this is a course I've recently created so I haven't finished adding all the sections that I want to add so in time I'm going to do that. Right now the page is accessible on the, you can find the course on the second page. And here it is, Trigonometry, the Unit Circle, Angles, and Right Triangles.
It's basically the one with the dark background and a circle with a triangle inside the circle. So let's look at the curriculum. In the first section, I'm going to go over angles, radians, how to convert degrees to radians, coterminal angles, how to convert DMS to decimal degrees, arc length, left area of the sector of a circle, linear speed and angular speed were problems.
And also, if you need to take the time that's shown on the clock, and if you need to convert it to an angle measure, I cover that in this section as well. And then at the end of each section is a video quiz. The next section is about the unicircle. The six trig functions, sine, cosine, tangent, secant, cosecant, cotan, and also reference angles as well. After that you have right triangle trigonometry, things like SOHCAHTOA, the special right triangles like the 30, 60, 90 triangle.
You need to know that so you can evaluate sine and cosine functions without using the unicircle. Next, we're going to talk about how to solve angle of elevation and depression problems, and just solving the missing sides of right triangles. After that, trigonometric functions of any angle, and then graphing trig functions. You need to know how to graph the sine and cosine functions, secant, cosecant.
and tangent as well. After that, inverse trig functions. You need to know how to evaluate it, and also how to graph it too. In addition, you need to know how to graph or evaluate composition of trig functions.
For example, you might have sine of inverse cosine of three of for something like that and you can use the right triangle to solve those types of problems you'll see when you are access that section after that applications of trig functions solving problems to have two right triangles in it and Barron's as well. One of the hardest sections in trig is this section, verifying trig identities. So that's a hard one, so make sure you spend some time learning that section. After that, sum and difference formulas, double angle, half angle, power reducing formulas, product to sum, sum to product, and also solving trig equations.
But there are still some sections I'm going to add to this course, like... For example, law of sines, law of cosines, polar coordinates, and some other topics as well. So about two-thirds of the course is finished so far. And for most students, this is just what they need in trig. But in time, you'll see more.
So now you know how to access the course, and if you have any questions, let me know. So thanks for watching.