SOLUTION: If α and β are acute angles such that csc (α) = 17/15 and cot (β) = 3/4, find the following. a. sin(α+β) b.tan(α+β)

Algebra ->  Trigonometry-basics -> SOLUTION: If α and β are acute angles such that csc (α) = 17/15 and cot (β) = 3/4, find the following. a. sin(α+β) b.tan(α+β)      Log On


   

Question 745825: If α and β are acute angles such that csc (α) = 17/15 and cot (β) = 3/4, find the following.
a. sin(α+β)
b.tan(α+β)
Answer by lwsshak3(11628) About Me  (Show Source):
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If α and β are acute angles such that csc (α) = 17/15 and cot (β) = 3/4, find the following.
a. sin(α+β)
b.tan(α+β)
***
use a & b for (α & β
csc(a)=17/15
sin(a)=15/17
cos%28a%29=sqrt%281-sin%5E2%28a%29%29=sqrt%281-%28225%2F289%29%29=sqrt%2864%2F289%29=8%2F17
tan%28a%29=sin%28a%29%2Fcos%28a%29=%2815%2F17%29%2F%288%2F17%29=15%2F8
...
cot(b)=3/4
tan(b)=4/3
opposite side=4
adjacent side=3
hypotenuse=5
sin(b)=4/5
cos(b)=3/5
..
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)=(15/17)(3/5)+(8/17)(4/5)=(45/85)+(32/85)=77/85
tan(a+b)=(tan(a)+tan(b))/(1-tan(a)*tan(b))
=[(15/8)+(4/3)]/[1-(15/8)(4/3)]
=[45/24+32/24]/[1-60/24]=77/24/-36/24=-77/36
..
Check with calculator:
sin(a)=17/15
a≈61.9275º
tan(b)=4/3
b≈53.1301
a+b=115.0526
sin(a+b)=sin(115.0526)≈0.9059
exact ans=77/85≈0.9059
tan(a+b)=tan(115.0526)=-2.139..
exact ans=-77/36≈-2.139..