Video Transcript
Simplify 31 sin squared 𝜃 plus 26
cos squared 𝜃.
In this question, we are asked to
simplify a trigonometric expression. This means we need to determine
which trigonometric identities to apply to simplify the given expression. We can do this by first considering
the given expression. We see that it involves a sum of
sine squared and cosine squared functions.
There are multiple different
identities we could use to rewrite this expression. For example, we could try the
double angle or angle sum or difference identities. However, when dealing with
expressions involving the sum of the square of both the sine and cosine functions,
we can easily simplify by using the Pythagorean identity. This tells us that for any angle
𝑥, the sum of the squares of the sin of 𝑥 and the cos of 𝑥 is equal to one.
We can use this identity to
simplify the expression. First, we will rewrite the
expression to apply this identity. We can split 31 sin squared of 𝜃
into five sin squared of 𝜃 plus 26 sin squared 𝜃. We can then take out a factor of 26
from the final two terms to obtain the following expression. We can now simplify the expression
by applying the Pythagorean identity to get five sin squared 𝜃 plus 26 times
one. We can now reorder these terms and
evaluate to find that 31 sin squared 𝜃 plus 26 cos squared 𝜃 simplifies to give 26
plus five sin squared 𝜃.
