Question Video: Solving Equations Involving an Inverse Trigonometric Function

Video Transcript

Solve the inverse sin of 𝑥 is equal to 𝜋 by four.

Now we read this as the inverse sin of 𝑥. So what does that actually mean? Well, the inverse is essentially the opposite. It’s the operation that undoes another operation. So the inverse sin of 𝑥 is the opposite operation to finding sin of something. Now, since sine and the inverse of sine are opposite operations of one another, if we take the sin of the inverse sin of 𝑥, we will just be left with 𝑥. And so we’re going to take the sin of both sides of this equation. As expected, that leaves us with 𝑥 on the left-hand side. And then on the right, we get sin of 𝜋 by four.

Now, we could use a calculator to perform this part of the calculation. However, this is one of the values that we should indeed know. We can use a table to help us remember the key trigonometric values we should know by heart. These are sin, cos, and tan of 𝜋 by six, 𝜋 by four, and 𝜋 by three radians. Now, of course, 𝜋 by six radians is equal to 30 degrees and 𝜋 by four radians is 45 degrees. 𝜋 by three radians is 60 degrees. And so this table of values also holds when we’re working with degrees.

To fill in this table, we write one, two, and three in the first row and then we reverse that in the second row. We then make all of these numbers into a fraction with a denominator of two as shown. Then we find the square root of all of our numerators. But of course, the square root of one is simply one, so we don’t actually need to write that.

And now we see that sin of 𝜋 by six is one-half and so on. Now to find the corresponding tan values, we divide the sine value by the cosine value. Since the denominators of the fractions are the same, this just looks like dividing the numerator. So tan of 𝜋 by six is one over root three. Then tan of 𝜋 by four is root two divided by root two, which is just one. And tan of 𝜋 by three is root three divided by one, which is just the square root of three.

Now remember, we said that 𝑥 was equal to sin of 𝜋 by four. sin is the first row and 𝜋 by four is the middle column. And so sin of 𝜋 by four must be equal to root two over two. And so that is our value for 𝑥.

The solution to the equation inverse sin of 𝑥 is equal to 𝜋 by four is 𝑥 equals root two over two.