Question Video: Using Odd and Even Identities to Evaluate a Trigonometric Function Involving Special Angles

Video Transcript

Find the value of tan of negative 𝜋 over four.

To solve this problem, we’re going to need to remember a few identities. We’re going to need to remember what the tan of negative 𝑥 is. The tan of negative 𝑥 is equal to the negative of tan of 𝑥. For us, that means that the tan of negative 𝜋 over four equals the negative of the tan of 𝜋 over four. At this point, we also need to recognise that 𝜋 over four is a common angle. 𝜋 over four is the radian representation of 45 degrees. And that means we’re considering the negative tan of 45 degrees.

Of course, you could’ve plugged in the beginning problem into a calculator. But let’s assume that you don’t have one. Let’s assume that you have to remember what the tan of 45 degrees is. The three common angles we need to consider are 30 degrees, 45 degrees, and 60 degrees. We need to consider the sine, cosine, and tangent. Beginning our chart, we write one, two, three and for the second row three, two, one. The threes and the twos have a square root. And everything has a denominator of two.

At this point, you might be wondering what the tangent would be. But to find the tangent, we take the numerator of sine and put it on top of the numerator of cosine. The tan of 30 degrees is one over the square root of three. The tan of 45 degrees is the square root of two over the square root of two, which we can rewrite as one. The tan of 60 degrees is the square root of three over one. And we’ll just write the square root of three. We were considering 45 degrees. The tan of 45 degrees is one.

If we plug in one for the tan of 45 degrees and bring down the negative, we can say that the value of tan of negative 𝜋 over four is negative one.