Meliton's trigonometric expression
The value of lo g 3 1 − tan ( 1 5 ∘ ) 1 + tan ( 1 5 ∘ ) can be expressed as b a , where a and b are coprime positive integers. What is the value of a + b ?
This problem is posed by Meliton C .
The answer is 3.
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3 solutions
Moderator note:
Nice! This was the approach that Meliton used, which simplifies the calculations needed.
We can use simply Goats peaceful in tan(15°)=2-√3 =1+2-√3/1-2+√3 =3-√3/-1+√3 =√3 =1/2=1+2=3
from previous solution, 1 − t a n x 1 + t a n x = c o s x − s i n x c o s x + s i n x
= c o s 2 x − s i n 2 x ( c o s x + s i n x ) 2
= c o s 2 x 1 + s i n 2 x
when x =15
1 − t a n 1 5 1 + t a n 1 5 = c o s 3 0 1 + s i n 3 0
=3/ 3
= 3
then l o g 3 3 = 1/2
a+b=3
It's not necessary if a/b is 1/2 then a+b would be 3.
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How?
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Simple ! 6/12 =1/2 but 6+12 is not 3 !
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@Srinivas Devulapalli – 6 & 12 are not co-prime. The question states a & b have to be co-prime. :-)
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@Sourav Majumdar – Indeed. The requirement of coprime positive integers in the question makes a and b uniquely determined.
Make sure to simplify your fractions when working with them.
Hint: Break tan into sine and cosine. Numerator will simplify to Cos(30) and denominator as 1-Sin(30).
Moderator note:
This hint is too cryptic, and it is not immediately clear how to proceed.
The approach suggests that we do not need to calculate tan 1 5 ∘ , but just the trig functions of 3 0 ∘ , which we know very well.
But what's the answer?
I think I have a better method- 1 can be written as tan45 in the numerator and in the denominator, write {1-tan15} as (1-tan45*tan15). This is the formula for tan(A+B). i.e. tan(45+15)=tan(60)=root three :)
tan(60) =tan45+tan15/(1-tan15*tan45) =(1+tan15)/(1-tan15)
log3(sqrt3)=1/2
a+b=1+2=3
Notice that, since tan 4 5 ∘ = 1 ,
1 − tan ( 1 5 ∘ ) 1 + tan ( 1 5 ∘ ) = 1 − 1 ⋅ tan ( 1 5 ∘ ) tan ( 4 5 ∘ ) + tan ( 1 5 ∘ ) = 1 − tan ( 4 5 ∘ ) tan ( 1 5 ∘ ) tan ( 4 5 ∘ ) + tan ( 1 5 ∘ ) = tan ( 4 5 ∘ + 1 5 ∘ ) , because of the Tangent Sum Identity. This yields a quick solution: 1 − tan ( 1 5 ∘ ) 1 + tan ( 1 5 ∘ ) = tan 6 0 ∘ = 3 = 3 1 / 2 . And therefore, a + b = 3 since lo g 3 3 1 / 2 = 2 1 .