Video: Solving Trigonometric Equations Using the Trigonometric Values of Special Angles

Given that sin 60° cos 30° − cos 60° sin 30° = sin 𝜃°, find the value of 𝜃 in degrees.


Video Transcript

Given that sin 60 degrees cos 30 degrees minus cos 60 degrees sin 30 degrees equals sin 𝜃, find the value of 𝜃 in degrees.

In order to solve this problem, we’re actually gonna have to use a trigonometric identity. And the identity we’re gonna use is that sin of 𝐴 minus 𝐵 is equal to sin 𝐴 cos 𝐵 minus cos 𝐴 sin 𝐵. But how does this relate to our problem?

Well, if we look back at our equation, we can see that actually we have 60 degrees would represent the 𝐴 in our identity. So we can say that 𝐴 is gonna be equal to 60 degrees. And then, we can also look at what 𝐵 would be. Well, in our example, 𝐵 would be equal to 30 degrees.

And we know that because our equation is actually in the same format as our identity because we have sin 60 cos 30, where they have sin 𝐴 cos 𝐵. Well, yes, 𝐴 is 60; 𝐵 is 30. So that works. And then, minus cos 60 sin 30, where we again said that 60 was 𝐴. So that’d be cos 𝐴, sin 𝐵 cause 𝐵 is 30. So great! That fits our identity.

So we now know that we can use this. And we can actually find the value of 𝜃 in degrees. So using our identity, we can see that sin 𝜃 is gonna be equal to sin of 𝐴 minus 𝐵. So now, we can substitute our values in, which means then therefore that sin 𝜃 is equal to sin of 60 minus 30. So therefore, we get that sin 𝜃 is equal to sin 30. So therefore, we can say that given that sin 60 cos 30 minus cos 60 sin 30 equals sin 𝜃, then therefore, 𝜃 is equal to 30 degrees.

But even that we’ve arrived at the answer, I always like to check to make sure that this is correct. So what I’d say is you can check it using a calculator. And if you put sin 60 cos 30 minus cos 60 sin 30 into your calculator, you’re gonna get 0.5.

But also remembering the quick tip that I’ve put there, do make sure that your calculator is in degrees. So in the display, there’s a little deg or it might just be a d because otherwise it might throw up a strange answer and you might think, oh! Have I got it wrong in the first place? But no, because the calculator is actually just on the wrong mode.

Okay, great! So we now know the answer of that first part. And we can now check by putting into our calculator sin 30. And when we do that, we also get 0.5. So therefore, we have our fully checked answer. And we can say with confidence that yes, 𝜃 is equal to 30 degrees.

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