Write out the sum formula for sin
<span>sin(x + y) = sinxcos + sinycosx </span>
<span>Then expand sin(a + b) + sin(a - b) </span>
<span>sinacosb + sinbcosa + sinacosb - sinbcosa </span>
<span>The 2nd and 4th terms cancel and you get </span>
<span>2sinacosb</span>
Sum/difference:
Let
This means that
Now, assume that 
is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get
if again we assume x to be rational, we have
But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.
- 4x from 8x = 4x
-9 + 19 = 4x -14
-9 + 19 = 10
10 = 4x -14
+14
24 = 4x
divide by 4
X = 6
<u>Direct Variation:</u> 
when y = 7: 
14 = 15x
<u>Inverse variation:</u> y*x = k
15 * 2 = k
30 = k
when y = 7: 7 * x = 30
Just so you know, you have to know at least one of the variables to find out what the other is. Like for example, X could equal 2, so that would make y equal 5. Or X could equal 23, and that would make y equal 26.
