Video Transcript
Find the set of values satisfying
tan 𝜃 squared plus tan 𝜃 equals zero, where 𝜃 is greater than or equal to zero
degrees and less than 360 degrees.
Upon inspection, we see that this
is a quadratic equation in tan 𝜃. Therefore, we’ll first factor the
left-hand side of our equation, which gives us tan 𝜃 times tan 𝜃 plus one equals
zero. Taking each of these factors and
setting them equal to zero gives us tan 𝜃 equals zero and tan 𝜃 plus one equals
zero. Subtracting one from both sides of
our second equation gives us the tan of 𝜃 equals negative one. We now have two solutions: the tan
of 𝜃 equals zero, or the tan of 𝜃 equals negative one.
Let’s recall the graph of a tangent
function. You could either use a calculator
or a computer to find a sketch of the graph 𝑦 equals tan of 𝑥. Inspecting the graph, we see that
the tangent function is equal to zero at zero degrees, 180 degrees, and also at 360
degrees. However, we’re only interested in
values of 𝜃 that is less than 360 degrees. This means we’ll exclude 𝜃 equal
to 360 degrees. For the interval 𝜃 is greater than
or equal to zero but less than 360, tan of 𝜃 equals zero at zero degrees and 180
degrees.
Next, we’ll consider the places
where tan of 𝜃 equals negative one. We see that this happens somewhere
between 90 degrees and 180 degrees and again somewhere between 270 degrees and 360
degrees. But to identify them more
accurately, we’ll need to use another method. We can take the tan inverse of both
sides of our second equation. 𝜃 is equal to the tan inverse of
negative one, which is negative 45 degrees.
Negative 45 degrees is outside our
interval for 𝜃, but we recall that the period of the tangent function is 180
degrees. And we can therefore say that the
tan of 𝜃 equals tan of 180 degrees plus 𝜃. 180 degrees plus negative 45
degrees equals 135 degrees. And our 135 degrees does fit with
what we see on the graph. And we want to do this process one
more time. We want to say that 𝜃 will also be
equal to 135 degrees plus 180 degrees, which equals 315 degrees. And that fits with the sketch of
tan of 𝑥 on the graph.
In the given interval, tan of 𝜃
equals zero at zero degrees and 180 degrees, and tan of 𝜃 equals negative one at
135 degrees and 315 degrees.
