In triangle ABC, a=9 and a(cos A)+b(cos C)+c(cos B)=450/29. Find the value of sin A.
cos(A)=((b^2+c^2-a^2)/(2bc)) etc.
So the equation can be written in terms of a,b,c and since a=9, just in terms of b and c.
Desmos shows an ellipse (eq4) which simplifies to
b^2-(49/29)bc-c^2-81=0 (eq5)
Solving for c (eq6) and reframing for y in terms of x (eq7)
The breakthrough is (eq8) which is the result of simplifying (b^2+c^2-a^2)/(2bc) using the a=9, b=x, and c as a function of b from the above. It appears to be constant!
(eq10), (eq11), (eq12) are just steps in simplifying. cos(A)=21/29
The last few equations are elementary solving for sin(A)=20/29
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Posted by Jer
on 2023-12-02 13:41:51 |
Solution (2nd time's a charm)
