Question Video: Calculating Using Exact Trigonometric Values Mathematics

Video Transcript

Given that sin 𝑥 cos 60 equals one-quarter, find the value of 𝑥, where 𝑥 is greater than zero and less than 90 degrees.

Now to answer this question, we might begin by spotting that cos of 60 is one of the trigonometric values that we should know by heart. And whilst we can derive these, we certainly don’t want to be doing that every time. So there’s a handy table that can help. This table contains values for sin, cos, and tan of zero, 30, 45, 60, and 90 degrees.

We fill this table in in the following manner. We begin by writing zero, one, two, three, and four. Then we reverse this for cosine: four, three, two, one, and zero. We add a denominator of two to every single one of these values. Then we take the square root of the numerator only. And this allows us to simplify a number of these expressions. The square root of zero is zero, so the square root of zero over two is also zero, meaning that sin of zero and cos of 90 are both zero.

The square root of one is simply one, so sin of 30 and cos of 60 are one-half. Finally, the square root of four is two. So sin of 90 and cos of zero are two over two, which is simply equal to one. So those are our values for sin of 𝜃 and cos of 𝜃.

But what about tan of 𝜃? Well, we divide any value for sin of 𝜃 by any value of cos of 𝜃 to get the corresponding value of tan of 𝜃. tan of zero then is zero divided by one, which is zero. tan of 30 is one-half divided by root three over two. In fact, though, since the denominators are the same, we can simply divide the numerators. So tan of 30 is one over root three, which is equivalent to root three over three.

tan of 45 is root two over two divided by root two over two, which is one. tan of 60 is root three divided by one, which is root three. And then when we try to find tan of 90, we get one divided by zero, which is undefined. So tan of 90 is undefined. And this is really useful because we can now rewrite the left-hand side of our equation by finding the value of cos 60 in our table. cos of 60 is equal to one-half. So our equation sin 𝑥 cos 60 equals a quarter becomes sin 𝑥 times one-half equals a quarter.

And in fact, it will be helpful to isolate sin of 𝑥 because then we can find the relevant values in our table. To do so, we’re going to divide by one-half. And actually, that’s the same as multiplying by two. So sin of 𝑥 is a quarter times two, which is simply equal to one-half. So we now need to ask ourselves what value of 𝑥 in our table gives us sin 𝑥 equals one-half. Well, we might notice that sin of 30 equals one-half. So for the equation sin 𝑥 equals one-half to hold for values of 𝑥 between zero and 90 degrees, 𝑥 must be equal to 30.

So given that sin 𝑥 cos 60 equals a quarter, our value of 𝑥 is 30 degrees.