SOLUTION: sin(x+y)=sin(x)cos(y)+sin(y)cos(x) 20 sin(x)+21 cos(x).

Algebra ->  Trigonometry-basics -> SOLUTION: sin(x+y)=sin(x)cos(y)+sin(y)cos(x) 20 sin(x)+21 cos(x).      Log On


   

Question 1090465: sin(x+y)=sin(x)cos(y)+sin(y)cos(x)
20 sin(x)+21 cos(x).
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!

20sin%28x%29%2B21cos%28x%29

We draw a right triangle with legs having the lengths of those
two coefficients.  We put the 20 on the adjacent side to y
and 21 on the opposite side, so that we can use the given 
sin(x+y) formula.  [If we did it the other way we'd have to 
use the cos(x-y) formula.]

 

Note here that y is the angle whose tangent is 21/20, and we
state that mathematically by the equation y = tan-1(21/20).

We calculate the hypotenuse by the Pythagorean theorem:

c%5E2=a%5E2%2Bb%5E2
c%5E2%2B20%5E2%2B21%5E2
c%5E2=400%2B441
c%5E2=841
c=sqrt%28841%29
c=29

 

We multiply the given expression, which is 

20sin%28x%29%2B21cos%28x%29

by 29%2A%281%2F29%29 which is the same as multiplying
by 1, and does not change the value:

29%281%2F29%29%2820sin%28x%29%5E%22%22%2B21cos%28x%29%29

We distribute the %281%2F29%29

29%28expr%2820%2F29%29sin%28x%29%2Bexpr%2821%2F29%29cos%28x%29%29

We use the right triangle above to substitute trig
ratios for the fractions:

29%28cos%28y%29%5E%22%22sin%28x%29%2Bsin%28y%29cos%28x%29%29

We rearrange the factors in the parentheses so that 
what's inside the parentheses will look like the right 
side of the given formula sin(x+y)=sin(x)cos(y)+sin(y)cos(x)

29%28sin%28x%29%5E%22%22cos%28y%29%2Bsin%28y%29cos%28x%29%29

We replace the parentheses using the given formula:

29sin%28x%2By%29, where y = tan-1(21/20).

Edwin