Video Transcript
Simplify the cos of 𝜃 plus the cos
of 180 degrees minus 𝜃.
There are a few ways of simplifying
this expression. For example, we could recall the
supplementary angle identity. That tells us that the cos of 180
degrees minus 𝜃 is equal to negative cos of 𝜃. Substituting this into our
expression, we would have the cos of 𝜃 plus negative cos of 𝜃. As this is the same as subtracting
cos of 𝜃 from cos of 𝜃, our answer would be zero.
Whilst this method appears
straightforward, it relies on us remembering the supplementary angle identity
shown. There are many such identities, and
trying to remember all these is difficult. It is therefore helpful to
understand where these identities come from using the unit circle. If we sketch an acute angle 𝜃 in
standard position, then the angle 180 degrees minus 𝜃 is the supplementary angle to
𝜃. Together, these form a straight
line on the 𝑥-axis as shown.
Reflecting our triangle in the
vertical axis, we have the angle 180 degrees minus 𝜃 in standard position. Since the triangles are congruent,
their bases will be equal in length. This leads us to the identity the
cos of 180 degrees minus 𝜃 is equal to the negative of cos of 𝜃. As this is true for any angle 𝜃
measured in degrees, we know that our answer is correct. cos of 𝜃 plus cos of 180 degrees
minus 𝜃 is equal to zero.
