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help finding value of tan and cos

 
 
1.  How do  I find the value of tan 15 degrees without using tables or calculators
 
 
2.  How do I find the value of cos 105 degrees without using tables or calculators
 
 
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1 Answer

1)  15 degrees = 45 degrees - 30 degrees, so use the tangent difference-of-two-angles identity:
 
                    tan(u)-tan(v)
tan(u-v) = -------------------
                   1+tan(u)tan(v)
 
where u = 45o and v = 30o.  We know that tan(45o) = 1 and tan(30o) = 1/√3.
 
2)  cos(105o) = cos(90o+15o) = -sin(15o).  Use the sine difference-of-two-angles identity:
 
-sin(45o-30o) = sin(30o)cos(45o)-sin(45o)cos(30o)    [Note that the - sign (-sin(15o)) reverses the usual order]
 
Where sin(45o)=cos(45o)=1/√2, sin(30o)=1/2, cos(30o)=√3/2.
 
[105o is also = 60o+45o, so you could also use the cosine sum-of-two-angles identity]