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sergiy2304 [10]
3 years ago
9

Please prove this........​

Mathematics
1 answer:
Crazy boy [7]3 years ago
7 0

Answer:  see proof below

<u>Step-by-step explanation:</u>

Given: A + B + C = π    →     C = π - (A + B)

                                    → sin C = sin(π - (A + B))       cos C = sin(π - (A + B))

                                    → sin C = sin (A + B)              cos C = - cos(A + B)

Use the following Sum to Product Identity:

sin A + sin B = 2 cos[(A + B)/2] · sin [(A - B)/2]

cos A + cos B = 2 cos[(A + B)/2] · cos [(A - B)/2]

Use the following Double Angle Identity:

sin 2A = 2 sin A · cos A

<u>Proof LHS → RHS</u>

LHS:                        (sin 2A + sin 2B) + sin 2C

\text{Sum to Product:}\qquad 2\sin\bigg(\dfrac{2A+2B}{2}\bigg)\cdot \cos \bigg(\dfrac{2A - 2B}{2}\bigg)-\sin 2C

\text{Double Angle:}\qquad 2\sin\bigg(\dfrac{2A+2B}{2}\bigg)\cdot \cos \bigg(\dfrac{2A - 2B}{2}\bigg)-2\sin C\cdot \cos C

\text{Simplify:}\qquad \qquad 2\sin (A + B)\cdot \cos (A - B)-2\sin C\cdot \cos C

\text{Given:}\qquad \qquad \quad 2\sin C\cdot \cos (A - B)+2\sin C\cdot \cos (A+B)

\text{Factor:}\qquad \qquad \qquad 2\sin C\cdot [\cos (A-B)+\cos (A+B)]

\text{Sum to Product:}\qquad 2\sin C\cdot 2\cos A\cdot \cos B

\text{Simplify:}\qquad \qquad 4\cos A\cdot \cos B \cdot \sin C

LHS = RHS: 4 cos A · cos B · sin C = 4 cos A · cos B · sin C    \checkmark

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  n = 8

Step-by-step explanation:

"By inspection" is an appropriate method.

We are asked to compare the expressions

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More formally, we could write ...

  n^2 = 8n . . . . the two formulas give the same value

  n^2 -8n = 0 . . . . rearrange to standard form

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Using the zero product rule, we know the solutions will be the values of n that make the factors zero. Those values are ...

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Generally, we're not interested in "trivial" solutions (n=0), so the only value of n that is of interest is n = 8.

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A recipe for sparkling grape juice calls for 1 1/2 quarts of sparkling water 3/4 quart of grape juice. Q1: How much sparkling wa
kogti [31]

Answer:

1) 18 Quart of Sparkling water need to mix with 9 quart grape juice

2) 15/2  Quart of grape juice required for 15/4 quarts of sparkling water.

3) Quantity of grape juice  and  sparkling water in 100 quart of punch are 33 1/3 quart  and 66 2/3 quart respectively.

Step-by-step explanation:

1 1/2 quarts = 1 + (1/2) = 3/2 quarts

1)  water required for 3/4 quart of grape juice =  3/2 quarts

so water required for 1 quart of grape juice = (3/2) ÷ (3/4) = (3/2)× (4/3) = 2 quarts

so water required for 9 quart of grape juice = 9 * 2 = 18 quart

18 Quart of Sparkling water need to mix with 9 quart grape juice

2) From solution of 1 ,

   For 18 quart of Sparkling water , grape juice require = 9 quart

   So for 1 quart of Sparkling water , grape juice require = 9÷18 = 1/2 Quart

   so for 15/4 quart of Sparkling water ,  grape juice require = 1/2 × 15/4 =

15/2  Quart of grape juice required for 15/4 quarts of sparkling water.

3)

in 1 we calculated that 18 Quart of Sparkling water need to mix with 9 quart grape juice that is 2 quart of Sparkling water need to mix with 1 quart of grape juice.

In other words 1 quart of grape juice + 2 quart of sparkling water is 3 quart of punch.

Quantity of Grape juice in 3 quart of punch = 1 quart

so quantity of grape juice in 1 quart of punch = 1/3 quart

And quantity of grape juice in 100 quart of punch = 100×(1/3) = 100/3 =

33 1/3

Quantity of sparkling water in 3 quart of punch = 2 quart

so quantity of sparkling water  in 1 quart of punch = 2/3 quart

And quantity of sparkling water  in 100 quart of punch = 100 ×2/3 quart = 66 2/3


 


     


3 0
3 years ago
Whats the radius and center of (x+4)^2+(y-2)^2=9
Fiesta28 [93]
I believe the center would be (-4,2) while the radius should be 3.
8 0
3 years ago
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