Find csc 𝛽 given 𝛼 and 𝛽 are two
complementary angles, where sec 𝛼 is equal to five-quarters.
We begin by recalling that two
complementary angles sum to 90 degrees. In this question, this means that
𝛼 and 𝛽 sum to 90 degrees. Subtracting 𝛼 from both sides of
this equation, we have 𝛽 is equal to 90 degrees minus 𝛼. We are trying to calculate csc of
𝛽. Therefore, this must be equal to
the csc of 90 degrees minus 𝛼. As we are told in the question that
the sec of 𝛼 is five-quarters, we’ll need to rewrite our expression using the
reciprocal and cofunction identities.
One of the cofunction identities
states that the sin of 90 degrees minus 𝜃 is equal to the cos of 𝜃. Since the cosecant and secant
functions are the reciprocal of the sine and cosine functions, respectively, we have
csc 𝜃 is equal to one over sin 𝜃 and sec 𝜃 is one over cos 𝜃. This means that the csc of 90
degrees minus 𝜃 is equal to the sec of 𝜃.
In this question, we have the csc
of 90 degrees minus 𝛼, which is equal to sec 𝛼. And as already mentioned, we are
told that this is equal to five-quarters. If 𝛼 and 𝛽 are two complementary
angles where the sec of 𝛼 is five-quarters, then the csc of 𝛽 must also equal