If sin (a) +cos (a) =1.2, then what is sin(a)^3 + cos(a)^3?
I was having some trouble with this problem. I got .98 as an estimate. Why is this wrong!!!1
Why not show how you arrived at that value?
If (sin+cos)=1.2,
(sin+cos)^2 = 1.44
sin^2+2sin*cos+cos^2 = 1.44
1+2sin*cos = 1.44
sin*cos = .22
since (x+y)^3 = (x+y)(x^2-xy+y^2)
(sin^3+cos^3) = (sin+cos)(sin^2-sin*cos+cos^2)
= (sin+cos)(1-sin*cos)
= 1.2 * 0.78
= 0.936
Thanks! Actually, I had like the same method as you for like the first few steps then I had a miscalculation
To find the value of sin(a)^3 + cos(a)^3, we can use the identity for the cube of a binomial: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
In this case, a = sin(a) and b = cos(a). We can rewrite the expression as follows:
sin(a)^3 + cos(a)^3 = (sin(a) + cos(a))(sin(a)^2 - sin(a)cos(a) + cos(a)^2).
Now, we are given that sin(a) + cos(a) = 1.2. Substituting this value into our expression, we get:
sin(a)^3 + cos(a)^3 = (1.2)(sin(a)^2 - sin(a)cos(a) + cos(a)^2).
To find the value of sin(a)^2 - sin(a)cos(a) + cos(a)^2, we can use a trigonometric identity:
sin(a)^2 + cos(a)^2 = 1.
Rearranging this equation, we get:
sin(a)^2 = 1 - cos(a)^2.
Substituting this into our expression, we have:
sin(a)^3 + cos(a)^3 = (1.2)((1 - cos(a)^2) - sin(a)cos(a) + cos(a)^2).
Simplifying further:
sin(a)^3 + cos(a)^3 = (1.2)(1 - sin(a)cos(a)).
Now, we need to determine the value of sin(a)cos(a). We can use another trigonometric identity:
sin(2a) = 2sin(a)cos(a).
Rearranging this equation, we get:
sin(a)cos(a) = (1/2)sin(2a).
Substituting this into our expression, we have:
sin(a)^3 + cos(a)^3 = (1.2)(1 - (1/2)sin(2a)).
To calculate sin(2a), we can use the double-angle identity for the sine function:
sin(2a) = 2sin(a)cos(a).
Given sin(a) + cos(a) = 1.2, we can square this equation to get:
(sin(a) + cos(a))^2 = (1.2)^2.
Expanding the square and using the identity sin^2(a) + cos^2(a) = 1, we have:
sin(a)^2 + 2sin(a)cos(a) + cos(a)^2 = 1.44.
Since sin(a)^2 + cos(a)^2 = 1, we can rewrite the equation as:
2sin(a)cos(a) = 1.44 - 1 = 0.44.
Now, substituting this value into our expression, we get:
sin(a)^3 + cos(a)^3 = (1.2)(1 - (1/2)sin(2a)) = (1.2)(1 - (1/2)(0.44)).
Simplifying:
sin(a)^3 + cos(a)^3 = (1.2)(1 - 0.22) = (1.2)(0.78) = 0.936.
Therefore, the correct value of sin(a)^3 + cos(a)^3 is approximately 0.936.
It seems that your estimate of 0.98 was close but not exactly correct. Perhaps there was a rounding error or a mistake in the calculations. By following the above steps and double-checking your calculations, you should be able to obtain the correct answer.