Trigonometry. How to calculate the unknown sides of a right angled triangle.

Welcome back, my viewers. Today we will be discussing this particular question, and this is under trigonometry. When you look at this diagram, it looks somehow Tricky, but it's a very simple thing to solve. Okay, we are asked to find the values of a and b. How do we go about this? First of all, you need to know the three basic trigonometric ratios which are sine, cosine, and tangent. Okay, sine theta is equal to opposite over hypotenuse. Your cosine theta is equal to adjacent over what? Hypotenuse. And your tan theta is equal to opposite over adjacent. So, because of this, there is this acronym that we use or that you can use to remember these ratios. Which are which? Which is which? And so on. Alright, so this is actually the sign. opposite over hypotenuse. This is what I've already stated. Then what is cos? Adjacent over hypotenuse. I've written that down, and tan is what? Opposite over adjacent. Okay, now what is The opposite? Which side of the triangle is opposite? This side of the triangle, sorry, On the contrary, it's the hypotenuse and the adjacent side. So let me introduce that to you. Basically. Okay, let me just list it. If you have a right-angled triangle, okay, then your right Angle, okay, the longest side is always your hypotenuse. Okay, it's always the longest side. or you say the side that is following, that is facing your, that is facing your what? Your right angle. Then, if this is your angle, the angle of reference, then the side opposite that angle is your opposite. Okay, that means the side facing your angle is your opposite. The remaining side is your adjacent. The remaining side is your adjacent. But note that opposite and adjacent Don't have a specific position. No, their position changes according to the angle's position. What do I mean? Notice that since this is the side facing our angle here, that means that this This side is the opposite, and this side is the adjacent. Now watch what happens if I change the position of this angle. You will notice that your opposite and adjacent sides will also change accordingly. That means If this is your right-angled triangle and this is this, your hypotenuse remains the same. It's always the longest side and the side that faces your right angle. Now if I were to decide that this is my reference angle now, if this is the position of my angle, then this means that this the side facing it becomes my opposite, and the remaining side becomes my adjacent. Can you see that? So your The opposite and your adjacent don't really have a particular base or a particular position assigned. to them. They change depending on what? The position of your angle. Alright, this is just an Introduction to what we are about to see here. We are going to use this knowledge that we just acquired. here to relate to what we have on the board. So look at the board very carefully and you will notice that We are asked to find A. To find A, this A now, I'm going to do something. I'm going to neglect I'm going to focus on this line here, and I'll concentrate on angle ABC. I'm going to focus on triangle ABC. So if I were to draw this triangle ABC, it would look like this. This is point A, This is our B, this is our C. I'm going to ignore this line. Okay, this is the A we are looking for. What is this? 25 degrees and this is 17 meters. Alright, finding this A is actually very easy. This is A. Finding this is very easy. So if we should, if we should bring out the parameters We have here, remember what I explained initially. It means that our hypotenuse, this is the longest. The side facing our angle is the opposite side. Our right angle is 17 meters. So our hypotenuse is equal to 17 meters. Then our opposite is the side facing the angle, right? This angle of reference, That's it. That means our opposite is A. That means the opposite is equal to A and our theta. What is it? 25 degrees. Now these are the three things we have right now. These are the three. things we have right now. So if we need to solve for our theta right now, sorry, if we need to To solve for our A, we are going to relate the opposite side to the hypotenuse. This is the opposite side, and this is the hypotenuse. hypotenuse. These are the sides we need to relate. Relate the side they are looking for. on the side that is giving. So opposite over hypotenuse, remember our so-ka-so-a. So-ka-so-a, Okay? So it means that sine is what? Opposite over hypotenuse. So notice that it is this sign that connects the opposite side to the hypotenuse. Opposite and hypotenuse. So because it is this The sign that connects the opposite side and the hypotenuse is what we will use: sine, to solve this. We are going to write that sine theta, that is sine 25, is going to be equal to our opposite, which is A. what we are looking for over our hypotenuse is what? 70. Now, if we cross-multiply, you notice That your A is what? 70 times sine 25, and this will give you 29.58 meters. So this What is the side? 29.58 meters. You can see how easy it is. As easy as A, B, C. So we are going to use the same idea to obtain the value of our B, except that our B can be obtained if we are able to get A, C. Notice that our B should be A, C minus what? A, D, right? The entire A, C minus A, D will give us our B. So how do we solve it now? We can actually solve it very easily. So now we can find A, C from this triangle. Our A, C is the other side, right? This is the opposite. is the hypotenuse, which means the remaining side is the other leg. Alright, if we can find A, C, then we are going to subtract 30 from this side from the entire length of this side, which is A, C, to obtain Our B. Alright, so let's go ahead and do that. Now, since we're looking for adjacent ones, right? Since we are looking for the adjacent side, which is A, C, we are going to connect it to the given side. which is our hypotenuse. Remember, the side we are looking for, which is the adjacent, We are going to connect it to the given side, which is the hypotenuse. So, excuse me, adjacent and hypotenuse are related by cosine, okay? So it means that cos 25 equals, Remember, cos is adjacent over hypotenuse—the adjacent side you're looking for over the hypotenuse side. That is given. So our adjacent side is A, C, the line A, C, divided by our hypotenuse, which is 70. This means that our A, C will be equal to 70 cos 25, and this will give us 63.44, 63.44 meters. So this means that from here to here is what? 63.44 meters. Alright, Now remember, we are not looking for A, C, but rather for B, C, that is B. And our B is what? But B is equal to, B is equal to A, C minus A, D, A, C minus A, D, Alright? Line A, C minus line A, D. Now what did we get for A, C? We got A, C as 69 point. Sorry, we got A, C as 63.44. So it's going to be 63.44 minus our A, D. Our A, D is 30 meters. minus 30, and when you do that, you are going to get what? 33.44 meters. It's going to be equal To D. Therefore, our A is equal to 29.58 meters, and our B is equal to 33.44 meters. Thank you. Thank you so much for watching. Have a lovely day and goodbye.