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).