Using the half angle formulas, or otherwise, find the exact value of the tan of 𝜋 by eight.
Before we are starting this question, we recall that 𝜋 radians is equal to 180 degrees. This means that 𝜋 over four radians is equal to 45 degrees. And we recall that this is one of our special angles. Both the sin and cos of 45 degrees are equal to root two over two, and the tan of 45 degrees is equal to one. We will use some of these values to help us solve this problem.
We are told to use the half angle formulas. We recall that the tan of 𝜃 over two is equal to one minus the cos of 𝜃 all divided by sin 𝜃 or the sin of 𝜃 divided by one plus the cos of 𝜃. In this question, we will consider the first version of the formula. The tan of 𝜋 over eight must therefore be equal to one minus the cos of 𝜋 over four divided by the sin of 𝜋 over four. Recalling that 𝜋 over four radians is 45 degrees, we can substitute sin of 𝜋 over four and the cos of 𝜋 over four with root two over two.
Splitting the numerator here gives us one over root two over two minus root two over two over root two over two. This is equal to two over root two minus one. We can rationalize the denominator of the first term by multiplying the numerator and denominator by root two. This gives us two root two over two which is equal to root two. The whole expression simplifies to root two minus one, which can also be written as negative one plus root two. This is the exact value of the tan of 𝜋 by eight radians.