Find the exact value of inverse tan of negative one in radians in the interval minus 𝜋 is less than 𝜃 which is less than or equal to two 𝜋 over three.
Before we even think about answering this question, it’s really important to notice that it asks us to give our answer as an exact solution. Now, if we are lucky enough to have a scientific calculator to hand, we can type inverse tan of negative one into this. And it should as long as your calculator is in radians give us an answer of negative 𝜋 over four.
If a calculator is not allowed however, we will need to recall the definition of an inverse function to help us. A function has an inverse function if and only if it is one to one, meaning that each 𝑦-value has no more than one corresponding 𝑥-value.
We can see that the tan graph fails this test. For any 𝑦-value on the tan graph, there are a whole number of corresponding 𝑥 solutions. Instead, we restrict the domain of our tan function to between negative 𝜋 over two and positive 𝜋 over two. On our unit circle, the values of inverse tan will be located on the right half of the circle, not including negative 𝜋 over two and positive 𝜋 over two since the tangent function is undefined at these points. These are the reference angles we know by heart.
We can then use the symmetry of the unit circle to find the corresponding values that lie in the fourth quadrant. Now, recall tan 𝜃 is equal to opposite over adjacent. In unit circle terms, tan 𝜃 is equal to 𝑦 over 𝑥.
We need to find an ordered pair on our unit circle that is between 𝜋 over two and negative 𝜋 over two such that 𝑦 over 𝑥 is equal to minus one. When 𝜃 equals negative 𝜋 over four, tan 𝜃 is equal to root two over two divided by negative root two over two. This is equal to negative one.
Therefore, the inverse tan of negative one is equal to negative 𝜋 over four.