Question 963649: Please solve this equation exactly.
sin(arccos(3/5) - arctan(5/12)) =
Found 3 solutions by lwsshak3, Edwin McCravy, ikleyn:
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Please solve this equation exactly.
sin(arccos(3/5) - arctan(5/12)) =
***
let reference angle x=arccos(3/5)
cosx=3/5
sinx=4/5
let reference angle y=arctan(5/12)
cosy=12/13
siny=5/13
Identity: sin(x-y)=sinxcosy-cosxsiny=4/5*12/13-3/5*5/13=48/65-15/65=33/65
sin(arccos(3/5) - arctan(5/12)) =33/65
..
Check:
arccos(3/5)≈53.13˚
arctan(5/12)≈22.62˚
53.13-22.62=30.51˚
sin(30.51)≈0.5077
exact value as computed above=33/65≈0.5077
Answer by Edwin McCravy(20086)  (Show Source):
You can put this solution on YOUR website!
A little more detail. It's good to draw triangles
on x and y axes for such problems:
sin(arccos(3/5) - arctan(5/12))
Let A = arccos(3/5) and B = arctan(5/12)
The we use the identity:
sin(A-B)= sin(A)cos(B)-cos(A)sin(B)
First we draw A = arccos(3/5).
arccos(3/5) means
"the angle in the first quadrant whose cosine is 3/5".
So we draw a triangle in the first quadrant. Since
cosine = adj/hyp = x/r we make the adjacent side, x, the
same as the numerator of 3/5, which is x=3 and make
the hypotenuse, r, the denominator of 3/5 which is r=5.
  
Next we draw arctan(5/12).
arctan(5/12) means
"the angle in the first quadrant whose tangent is 5/12".
So we draw another right triangle in the first quadrant.
Since tangent = opp/adj = y/x we make the opposite side, y, the
same as the numerator of 5/12, which is y=5 and make
the adjacent side, x, the denominator of 5/12 which is x=12.
  
Now we can easily finish:
sin(A-B)= sin(A)cos(B)-cos(A)sin(B)
= (4/5)(12/13)-(3/5)(5/13)
= 48/65 - 15/65
= 33/65
Edwin
Answer by ikleyn(53875)  (Show Source):
You can put this solution on YOUR website! .
Trigonometry-basics/963649: Please solve this equation exactly.
sin(arccos(3/5) - arctan(5/12)) =
~~~~~~~~~~~~~~~~~~~~~~~~~~
As worded/printed/presented in the post, the problem makes no sense and can not be solved/answered as requested.
The first term is a dimensionless real rational number, while the second term is the angle measure (in radians or in degrees).
So, it makes no sense to perform arithmetic operations on these numbers/expressions.
To the reader: ignore this post, since it is nonsense.
Ignore the "solution" by @lwsshak3, since it is nonsense, too.
Keep in mind, that the computer code, which @lwsshar3 uses in his "solutions", is not able
to distinct nonsense from sense, so it simply deceives you in this case, instead of making an accurate analysis.
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