suppose that sin(x) + cos(x) = 4/3
a)square this result and find a value for sin(x)cos(x).
b)using the result from a) find the exact value of sin^3(x)+cos^3(x). (Hint: Do you recall how to factor x^3 + y^3)
To solve part a), we need to square the equation sin(x) + cos(x) = 4/3 and find the value of sin(x)cos(x).
Let's start by squaring the equation:
(sin(x) + cos(x))^2 = (4/3)^2
Expanding (sin(x) + cos(x))^2:
(sin(x))^2 + 2sin(x)cos(x) + (cos(x))^2 = 16/9
Now, notice that we have sin(x)cos(x) in the equation. To find its value, let's rearrange the equation:
2sin(x)cos(x) = (16/9) - (sin(x))^2 - (cos(x))^2
Since sin^2(x) + cos^2(x) = 1 (from the Pythagorean identity), we can substitute it into the equation:
2sin(x)cos(x) = (16/9) - 1
Simplifying:
2sin(x)cos(x) = 7/9
Therefore, the value of sin(x)cos(x) is 7/18.
Moving on to part b), we need to find the exact value of sin^3(x) + cos^3(x) using the result from part a).
Recall that x^3 + y^3 can be factored as (x + y)(x^2 - xy + y^2). In our case, x represents sin(x) and y represents cos(x).
So, substituting in:
sin^3(x) + cos^3(x) = (sin(x) + cos(x))(sin^2(x) - sin(x)cos(x) + cos^2(x))
Using the result from part a) of sin(x)cos(x) = 7/18, we can substitute it into the equation:
sin^3(x) + cos^3(x) = (sin(x) + cos(x))(sin^2(x) - (7/18) + cos^2(x))
Remember that sin^2(x) + cos^2(x) = 1, so we can substitute it in as well:
sin^3(x) + cos^3(x) = (sin(x) + cos(x))(1 - (7/18))
Simplifying:
sin^3(x) + cos^3(x) = (sin(x) + cos(x))(11/18)
And that is the exact value of sin^3(x) + cos^3(x), using the value of sin(x)cos(x) obtained in part a).