Find sin 𝜃 plus cos 𝜃 squared
given that sin 𝜃 cos 𝜃 equals six over seven.
The first thing we can do with this
question is to write out sin 𝜃 plus cos 𝜃 squared as two sets of brackets. We can treat this like any other
binomial expansion and multiply out these brackets to find the following four terms:
sin 𝜃 times sin 𝜃 plus sin 𝜃 times cos 𝜃 plus cos 𝜃 times sin 𝜃 and plus cos
𝜃 times cos 𝜃.
Now, we know that the order in
which sin 𝜃 and cos 𝜃 are multiplied together does not matter. This means that cos 𝜃 sin 𝜃 is
the same as sin 𝜃 cos 𝜃. Upon realizing this, we see that
our two middle terms are in fact the same. And we can, therefore, group them
Another thing we can do to tidy up
our equation, is to see that sin 𝜃 times sin 𝜃 is sin 𝜃 squared. This can equivalently be written as
sin squared 𝜃, the same being true for cos. Let’s rewrite these two terms in
our expression. Now that we’ve tidied things up,
let’s rearrange our terms.
Looking at the first two terms of
our expression, you may recognize one of the Pythagorean identities. This identity states that sin
squared 𝜃 plus cos squared 𝜃 is equal to one. The second thing we may notice is
that our question gives us the value for sin 𝜃 cos 𝜃.
We can, therefore, perform two
substitutions on our expression. First, we can replace sin squared
𝜃 plus cos squared 𝜃 with one. Next, we can replace sin 𝜃 cos 𝜃
with six over seven.
To make our addition easier, we can
rewrite one as seven over seven so that our denominators match. Multiplying six over seven by two,
we find 12 over seven. Adding seven over seven to 12 over
seven, we find our answer is 19 over seven.
We can therefore, say that sin 𝜃
plus cos 𝜃 all squared is 19 over seven when sin 𝜃 times cos 𝜃 is six over