Find the value of eight over sin 𝜃
times negative five over csc 𝜃.
In this expression, we have a trig
function and a reciprocal trig function. One strategy for evaluating a
trigonometric expression is to write it in terms of sine and cosine functions. The first term in this expression
is already in terms of sine and cosine. And to rewrite csc 𝜃 in terms of
sine or cosine, we recall that csc 𝜃 is the reciprocal of sin 𝜃 and that csc 𝜃
equals one over sin 𝜃.
Therefore, in the denominator of
this second term, we replace csc 𝜃 with one over sin 𝜃. If we think about negative five
over one over sin 𝜃, this is negative five divided by one over sin 𝜃. And dividing by a fraction is the
same as multiplying by the reciprocal of that fraction. Therefore, we can rewrite negative
five over one over sin 𝜃 as negative five times sin of 𝜃. In this case, we have a sin 𝜃 in
the denominator and a sin 𝜃 in the numerator, which cancels out, leaving us with
eight times negative five, which is negative 40.