help finding value of tan and cos

Question

 

 

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

 

 

thank you

Answer 1

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 = 45^o and v = 30^o.  We know that tan(45^o) = 1 and tan(30^o) = 1/√3.

 

2)  cos(105^o) = cos(90^o+15^o) = -sin(15^o).  Use the sine difference-of-two-angles identity:

 

-sin(45^o-30^o) = sin(30^o)cos(45^o)-sin(45^o)cos(30^o)    [Note that the - sign (-sin(15^o)) reverses the usual order]

 

Where sin(45^o)=cos(45^o)=1/√2, sin(30^o)=1/2, cos(30^o)=√3/2.

 

[105^o is also = 60^o+45^o, so you could also use the cosine sum-of-two-angles identity]