Video Transcript
Find the value of csc 56 degrees
divided by sec 34 degrees plus csc of 180 degrees minus 𝜃, given sin 𝜃 is equal to
one.
In order to answer this question,
we need to recall some of our key trigonometric identities. Firstly, the cosecant and secant
functions are the reciprocal of the sine and cosine functions, respectively, such
that csc 𝜃 is equal to one over sin 𝜃 and sec 𝜃 is equal to one over cos 𝜃. This means we can rewrite csc 56
degrees as one over sin 56 degrees. The sec of 34 degrees is equal to
one over the cos of 34 degrees. And the csc of 180 degrees minus 𝜃
is equal to one over sin of 180 degrees minus 𝜃.
We recall that dividing the
fraction 𝑎 over 𝑏 by the fraction 𝑐 over 𝑑 is the same as multiplying the first
fraction by the reciprocal of the second fraction. This means that one over sin of 56
degrees divided by one over cos of 34 degrees is equal to the cos of 34 degrees
divided by the sin of 56 degrees.
Our next step is to recall one of
our cofunction identities. The cos of 90 degrees minus 𝜃 is
equal to the sin of 𝜃. Since 34 degrees is equal to 90
degrees minus 56 degrees, then the cos of 34 degrees is equal to the sin of 56
degrees. The first part of our expression
simplifies to sin of 56 degrees over sin of 56 degrees, and this is equal to
one.
Next, let’s consider how we can
rewrite the sin of 180 degrees minus 𝜃. Recalling our CAST diagram, we know
that the sine of any angle between zero and 180 degrees is positive. And using our knowledge of the unit
circle, for any angle 𝜃, sin of 180 degrees minus 𝜃 is equal to sin 𝜃. This means that we can rewrite the
second part of our expression as one over sin 𝜃. We are told in the question that
sin 𝜃 is equal to one. This means that the entire
expression simplifies to one plus one, which is equal to two.
We can therefore conclude that the
value of csc 56 degrees over sec 34 degrees plus csc of 180 degrees minus 𝜃 is two
when sin 𝜃 equals one.
