Video Transcript
Simplify tan of 180 degrees minus
𝜃.
The tangent function has many
useful identities we can use to help us simplify expressions. However, we may not have memorized
these identities directly. So let us explore how to answer
this question by making use of the unit circle.
Recall that if we consider a point
on a unit circle, which forms an angle of 𝜃 with the positive 𝑥-axis, then its
𝑥-coordinate will be cos of 𝜃 and its 𝑦-coordinate will be sin of 𝜃. Thus, the coordinates of the point
can be written as cos 𝜃, sin 𝜃.
Now, what if we consider the angle
180 degrees minus 𝜃? Well, we know that if we go 180
degrees around a unit circle, we will end up pointing in the negative
𝑥-direction. This means that if we consider 180
degrees minus 𝜃, this will be an angle which is 𝜃 degrees clockwise from the
negative 𝑥-axis. In fact, we can see that this is
actually equivalent to reflecting the first point in the 𝑦-axis. Now, because this point also lies
on the unit circle, it will have coordinates cos of 180 minus 𝜃 and sin of 180
minus 𝜃. But we can also rewrite these
coordinates by comparing them to the first set of coordinates and using the symmetry
of the unit circle.
In particular, we can see that the
𝑥-coordinate is equal to negative cos 𝜃 because it is the same length as cos 𝜃
but in the opposite direction. As for the 𝑦-coordinate, we can
see that this is equal to sin 𝜃 because its distance from the 𝑦-axis is the
same. Therefore, the coordinates are
equal to negative cos 𝜃, sin 𝜃.
Using this information, we can
write out the supplementary angle trigonometric identities. First, by considering the
𝑥-coordinate of the point on the left, we can see that cos of 180 degrees minus 𝜃
equals negative cos 𝜃. Similarly, by considering the
𝑦-coordinates of this point, we find that sin of 180 degrees minus 𝜃 equals sin
𝜃.
Now, how can we relate these
identities to the tangent function? Recall that tan 𝜃 is equal to sin
𝜃 over cos 𝜃. Therefore, if we consider tan of
180 degrees minus 𝜃, we must have sine of that same angle over cosine of that same
angle. Then, by using the identities
above, we get sin 𝜃 over negative cos 𝜃. And finally, this can be simplified
to our final answer, negative tan 𝜃.
