SOLUTION: Evaluate the given expression without using a calculator. (Assume any variables represent positive numbers) Sin( arccos 3/5 + arctan 1/2)

Algebra ->  Trigonometry-basics -> SOLUTION: Evaluate the given expression without using a calculator. (Assume any variables represent positive numbers) Sin( arccos 3/5 + arctan 1/2)      Log On


   

Question 744764: Evaluate the given expression without using a calculator. (Assume any variables represent positive numbers)
Sin( arccos 3/5 + arctan 1/2)
Answer by lwsshak3(11628) About Me  (Show Source):
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Evaluate the given expression without using a calculator. (Assume any variables represent positive numbers)
Sin( arccos 3/5 + arctan 1/2)
***
let x=arccos 3/5
let y=arctan 1/2
..
cos%28x%29=3%2F5
sin%28x%29=sqrt%281-cos%5E2%28x%29%29=sqrt%281-9%2F25%29=sqrt%2816%2F25%29=4%2F5
..
tan%28y%29=1%2F2=sin%28y%29%2Fcos%28y%29
sin%28y%29=1%2Fsqrt%285%29
cos%28y%29=2%2Fsqrt%285%29
..
Sin( arccos 3/5 + arctan 1/2)=sin(x+y)
=sin(x) cos(y)+cos(x) sin(y)
=4/5*2/√5+3/5*1/√5
=8/5√5+3/5√5=11/(5√5)
=(11√5)/25
..
Check: (with calculator)
cos(x)=3/5
x≈53.13º
tan(y)=1/2
y≈26.57º
x+y≈79.7º
sin(x+y)=sin79.7º≈0.9839
exact answer=(11√5)/25≈0.9839