The sum of two numbers is 8. If one of them is x and the product is y, then the function expression of Y and X is (). The maximum product of these two numbers can be ()

The sum of two numbers is 8. If one of them is x and the product is y, then the function expression of Y and X is (). The maximum product of these two numbers can be ()

The functional expression of Y and X is (y = x (8-x)), and the maximum product of these two numbers can be (16)
y=x(8-x)=-x²+8x=-(x²-8x+16)+16=-(x-4)²+16

Derivation process of sum difference product formula in Mathematics

Sum difference product of sine and cosine
sin α+sinβ=2sin[(α+β)/2]·cos[(α-β)/2]
sin α-sin β=2cos[(α+β)/2]·sin[(α-β)/2]
cos α+cos β=2cos[(α+β)/2]·cos[(α-β)/2]
cos α-cos β=-2sin[(α+β)/2]·sin[(α-β)/2]
Method 1 the proof process of sin α + sin β = 2Sin [(α + β) / 2] · cos [(α - β) / 2]
Because
　　sin(α+β)=sin αcos β+cos αsin β,
　　sin(α-β)=sin αcos β-cos αsin β,
Add the left and right sides of the above two expressions to get the
　　sin(α+β)+sin(α-β)=2sin αcos β,
Let α + β = θ, α - β = φ
So
　　α=(θ+φ)/2,β=(θ-φ)/2
Substituting the values of α and β, we get
　　sin θ+sin φ=2sin[(θ+φ)/2]cos[(θ-φ）/2]
Method 2
According to Euler formula, e ^ IX = cosx + isinx
Let x = a + B
E ^ I (a + b) = e ^ ia * e ^ IB = (COSA + isina) (CoSb + isinb) = cosacosb sinasinb + I (sinacosb + sinbcosa) = cos (a + b) + isin (a + b)
So cos (a + b) = cosacosb sinasinb
　　sin(a+b)=sinacosb+sinbcosa
Pithy formula
Plus plus plus plus plus plus plus plus plus plus plus plus plus plus plus plus plus plus
Positive minus positive, remainder before, remainder minus remainder, negative sine
And vice versa
Look at the encyclopedia,
Sum difference product of tangent
Tan α ± Tan β = sin (α ± β) / (COS α · cos β) (with proof)
cotα±cotβ=±sin(β±α)/(sinα·sinβ)
tanα+cotβ=cos(α-β)/(cosα·sinβ)
tanα-cotβ=-cos(α+β)/(cosα·sinβ)
It is proved that the left side = Tan α ± Tan β = sin α / cos α ± sin β / cos β
　　=(sinα·cosβ±cosα·sinβ)/(cosα·cosβ)
= sin (α ± β) / (COS α · cos β) = right
The equation holds