Answer: see proof below
<u>Step-by-step explanation:</u>
Given: A + B + C = π → A = π - (B + C)
→ B = π - (A + C)
→ C = π - (A + B)
Use Sum to Product Identity: sin A - sin B = 2 cos [(A + B)/2] · sin [(A - B)/2]
Use the following Cofunction Identity: cos (π/2 - A) = sin A
<u>Proof LHS → RHS:</u>
LHS: sin A - sin B + sin C
= (sin A - sin B) + sin C
![\text{Factor:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{C}{2}\bigg)\bigg]](https://tex.z-dn.net/?f=%5Ctext%7BFactor%3A%7D%5Cqquad%202%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5B%20%5Csin%20%5Cbigg%28%5Cdfrac%7BA-B%7D%7B2%7D%5Cbigg%29%2B%5Ccos%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5D)
![\text{Given:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{\pi -(A+B)}{2}\bigg)\bigg]\\\\\\.\qquad \qquad =2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\cos \bigg(\dfrac{\pi}{2} -\dfrac{(A+B)}{2}\bigg)\bigg]](https://tex.z-dn.net/?f=%5Ctext%7BGiven%3A%7D%5Cqquad%202%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5B%20%5Csin%20%5Cbigg%28%5Cdfrac%7BA-B%7D%7B2%7D%5Cbigg%29%2B%5Ccos%20%5Cbigg%28%5Cdfrac%7B%5Cpi%20-%28A%2BB%29%7D%7B2%7D%5Cbigg%29%5Cbigg%5D%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cqquad%20%3D2%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5B%20%5Csin%20%5Cbigg%28%5Cdfrac%7BA-B%7D%7B2%7D%5Cbigg%29%2B%5Ccos%20%5Cbigg%28%5Cdfrac%7B%5Cpi%7D%7B2%7D%20-%5Cdfrac%7B%28A%2BB%29%7D%7B2%7D%5Cbigg%29%5Cbigg%5D)
![\text{Cofunction:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ \sin \bigg(\dfrac{A-B}{2}\bigg)+\sin \bigg(\dfrac{A+B}{2}\bigg)\bigg]](https://tex.z-dn.net/?f=%5Ctext%7BCofunction%3A%7D%5Cqquad%202%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5B%20%5Csin%20%5Cbigg%28%5Cdfrac%7BA-B%7D%7B2%7D%5Cbigg%29%2B%5Csin%20%5Cbigg%28%5Cdfrac%7BA%2BB%7D%7B2%7D%5Cbigg%29%5Cbigg%5D)
![\text{Sum to Product:}\qquad 2\sin \bigg(\dfrac{C}{2}\bigg)\bigg[ 2\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\bigg]\\\\\\.\qquad \qquad \qquad \qquad =4\sin \bigg(\dfrac{A}{2}\bigg)\cdot \cos \bigg(\dfrac{B}{2}\bigg)\cdot \sin \bigg(\dfrac{C}{2}\bigg)](https://tex.z-dn.net/?f=%5Ctext%7BSum%20to%20Product%3A%7D%5Cqquad%202%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29%5Cbigg%5B%202%5Csin%20%5Cbigg%28%5Cdfrac%7BA%7D%7B2%7D%5Cbigg%29%5Ccdot%20%5Ccos%20%5Cbigg%28%5Cdfrac%7BB%7D%7B2%7D%5Cbigg%29%5Cbigg%5D%5C%5C%5C%5C%5C%5C.%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%3D4%5Csin%20%5Cbigg%28%5Cdfrac%7BA%7D%7B2%7D%5Cbigg%29%5Ccdot%20%5Ccos%20%5Cbigg%28%5Cdfrac%7BB%7D%7B2%7D%5Cbigg%29%5Ccdot%20%5Csin%20%5Cbigg%28%5Cdfrac%7BC%7D%7B2%7D%5Cbigg%29)
64. Susan tipped at the rate of 75 dollars to 5 waiters.
Question: She can afford to pay up to 90 dollars, How many waiters can she tipped if that is the case.
Let’s solve first and identify how much will she be giving to each waitress with 75 dollars is to 5 waiters rate:
=> 75 dollars / 5 waiters = 15 dollars each waiter
Now, she have 90 dollars
=> 90 – 75 = 15
Thus
=> 90 / 15 = 6 waiters she will be tipping.
<span> 8 - 4x=4 - 3(2x+6)
<=> 8 - 4x = 4 - 6x - 18
<=> 8 - 4 + 18 = 4x - 6x
<=> 22 = -2x
x = -11</span>
