Find cos 𝜃 given sin 𝜃 is equal to negative three over five, where 𝜃 is greater than or equal to 270 degrees, but less than 360 degrees.
To start solving this problem, I am actually gonna use a little memory aid as you may have seen, which is SOH CAH TOA. And we’re actually gonna look at the first part. So cause what this actually tells us is that the sine of an angle or sine of 𝜃 is equal to the opposite divided by the hypotenuse. So this is in a right-angled triangle. It’s telling us that the actual values you’d have, i.e. the sine of 𝜃, is equal to opposite divided by hypotenuse.
So therefore, we can say that if sine 𝜃 is equal to negative three over five, that would mean that our opposite is going to be equal to three and our hypotenuse is going to be equal to five. There is also the negative sign to consider. But we’re gonna have a look at that a bit later on in the solution.
And now draw a little sketch to actually show what this value tells us. So if we’ve got a right-angled triangle, we can see that the opposite is three and the hypotenuse is five. But what can we do to find the adjacent? Well, what we can actually do is use the Pythagorean theorem cause we can say that A is going to be equal to the square root of five squared because that’s our hypotenuse squared minus three squared cause that’s one of our shorter sides squared. And we get that from a rearrange version of 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where we have the 𝑎 squared is equal to 𝑐 squared minus 𝑏 squared, where 𝑐 is the hypotenuse.
Okay, great! So we can solve this and find A. So we get that A is equal to root 16. So therefore A is equal to four. Now, the reason we wanted to find the adjacent side is because this second part of our SOH CAH TOA. It’s the CAH because what we’re trying to do is find cos. So we want to use CAH, which tells us that the cosine of 𝜃 or any angle is equal to the adjacent over the hypotenuse.
Okay, but we can now use this because we have the adjacent, which we’ve just found. And we already know the hypotenuse. So we can say therefore cos 𝜃 is equal to A over H. So then, we can say therefore cos 𝜃 is equal to four-fifths. But is this the final answer? So I want to draw your attention back to the negative sign we looked at earlier. sin 𝜃 was equal to negative three-fifths. So therefore, is cos 𝜃 can be equal to four-fifths or negative four-fifths?
And we’re gonna have a little look at the cos diagram to explain which one is the answer. Well, I’ve drawn a sketch of the cos diagram. And what this actually tells us is something really important. It tells us where our sine, cosine, or tangent of an angle are positive and where they’ll be negative. And the A stands for all. So therefore, we can say that all values between nought and 90 degrees are going to be positive. We can say that values between 90 and 180 are only going to be positive when it’s sin 𝜃. We can say between 180 and 270 degrees, sine will be positive if it’s tangent 𝜃. And we can say that only cos 𝜃 will gonna be positive between 270 and 360 degrees.
Well, if we look at the original question, we can see that’s actually this final section, the 270 degrees to 360 degrees, that we’re actually interested in. So we can see from the cos diagram that actually within this sector, the cosine of 𝜃 would be positive. And actually what we’ve said is that the sin 𝜃 was equal to negative three over five? Well, yes, that would be correct cause looking at the cos diagram would only be the top left section, where sin 𝜃 would be positive. So therefore, we can say that cos 𝜃 is equal to four-fifths.