Find the value of sin 𝜃 cos 𝜃
given sin 𝜃 plus cos 𝜃 is equal to five-quarters.
In order to solve this problem, we
will need to recall the Pythagorean identities. We will begin with the equation sin
𝜃 plus cos 𝜃 is equal to five over four. We can square both sides of this
equation. The left-hand side becomes sin 𝜃
plus cos 𝜃 multiplied by sin 𝜃 plus cos 𝜃. The right-hand side is equal to 25
over 16 as we simply square the numerator and denominator separately.
Distributing the parentheses or
expanding the brackets using the FOIL methods, we get sin squared 𝜃 plus sin 𝜃 cos
𝜃 plus sin 𝜃 cos 𝜃 plus cos squared 𝜃. We can group or collect the middle
terms. sin squared 𝜃 plus two sin 𝜃 cos
𝜃 plus cos squared 𝜃 is equal to 25 over 16. One of the Pythagorean identities
states that sin squared 𝜃 plus cos squared 𝜃 is equal to one. This means we can rewrite the
left-hand side of our equation as two sin 𝜃 cos 𝜃 plus one. We can then subtract one from both
sides of this equation. 25 over 16 minus one is equal to
nine over 16.
We can then divide both sides of
this new equation by two, giving us sin 𝜃 cos 𝜃 is equal to nine over 32. If sin 𝜃 plus cos 𝜃 is equal to
five over four, then sin 𝜃 multiplied by cos 𝜃 is equal to nine over 32.