Video Transcript
Two perpendicular forces of magnitudes 88 newtons and 44 newtons act at a point. Their resultant makes an angle 𝜃 with the 88-newton force. Find the value of sin 𝜃.
Well, the key here is this word “perpendicular” cause we’re told that the two forces are perpendicular to each other. So therefore, what we can do is use this to draw a sketch. So we’ve got a sketch here of our 44 newtons and 88 newtons. And as you can see, they are perpendicular to each other cause they’re at right angles to each other. And we’re also told that their resultant makes an angle 𝜃 with the 88-newton force. So we’ve also put this on our sketch.
Now to help us work out what the value of sin 𝜃 is, what you might like to do is just add this other side in. So we can see that, actually, what we’ve got here is a right-angled triangle. And we’ve got our 𝜃 within this. And then we’ve got two sides; they’re 88 newtons and 44 newtons. Well, if we remember our SOHCAHTOA, which is a memory aid we use to remember our trigonometric ratios, we can see that sin 𝜃 is gonna be equal to the opposite over the hypotenuse.
Well, if we take a look at our diagram, we can see that the opposite is 44 newtons. However, the hypotenuse we don’t know. So the first thing we need to do is to work out the hypotenuse. And the hypotenuse is, in fact, our resultant. And to do that, what we can do is use the Pythagorean theorem. And what this tells us is that if we’ve got the longest side, our hypotenuse, is 𝑐 and the other two sides are 𝑎 and 𝑏, then we can say that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. So therefore, we can say that 𝑅 squared is gonna be equal to 44 squared plus 88 squared. So therefore, 𝑅 squared is gonna be equal to root 9680.
So then if we take the square root of both sides, what we’re gonna get is that 𝑅 is equal to 44 root five. Now, we’re not interested in the negative value because we’re just looking in the magnitude of 𝑅. So it’s just going to be the positive values. So we’ve got 44 root five. So therefore, now what we can do is substitute in our opposite and our hypotenuse. So we’re gonna get sin 𝜃 is equal to 44, which is our opposite over our hypotenuse, which is 44 root five. So then what we’ve got is a common factor on the numerator and denominator. So we can divide through by 44. So we get sin 𝜃 is equal to one over root five.
Now, as we know, if we’ve got one over root five, if we ever have a surd as the denominator, then what we want to do is rationalize that denominator. And we’re gonna do that by multiplying one over root five by root five over root five. Well, the reason we multiply by root five over root five is because if we multiply root five and root five, what we get is five. And that’s because if we multiply two of the same root, we get the value that’s inside the root. And if we think about why that might be, if we had root five multiplied by root five, then this’d be the same as root 25. Well, root 25 is just five.
So, great, once we’ve done that, what we’re gonna get is root five over five. So therefore, we can say that if two perpendicular forces of magnitudes 88 newtons and 44 newtons act at a point and their resultant makes an angle 𝜃 with the 88-newton force, then the value of sin 𝜃 is root five over five.
