Could anyone tell me how to find the exact value of the tan 15 degrees?

Answer 1

Example: What’s the exact value of tan 15° or tan(π/12)?

Solution: You have to convert everything to radians, to make things simpler.

15° = 60°−45° (π/12 = π/3 − π/4). Therefore

tan(π/12) = tan(π/3 − π/4)

tan(π/12) = [tan(π/3) − tan(π/4)] / [1 + tan(π/3) tan(π/4)]

tan(π/12) = [(√3)− 1] / [1 + (√3)×1]

tan(π/12) = [(√3)− 1] / [(√3) + 1]

If you like, you can rationalize the denominator:

tan(π/12) = [(√3)− 1]² / [(√3) + 1]×[(√3) − 1]

tan(π/12) = [3 − 2(√3) + 1] / [3 − 1]

tan(π/12) = [4 − 2(√3)] / 2

tan(π/12) = 2 − √3

Answer 2

Graphically, start with a 30-60-90 triangle, with sides equal to 1, √3, and 2. Now extend the longer leg the same length as the hypotenuse. This gives you a 15-75-90 triangle. The shorter leg is still 1. The longer leg is 2 + √3. So, the tangent of 15° is 1/(2 + √3) = 2 - √3, about .268.

Answer 3

It's pretty complex and hard to do, and it derives from calculus, but tangent can be approximated by the Taylor Series.

tan n = summation notation from 1 to infinity of
[ (Bsub2n)(-4)^n(1-4^n) ] / 2n! * x^(2n-1)
B is a Bernoulli number. I found this from wikipedia so I'm not sure of what a Bernoulli number means, but well for the Summation Notation you just plug in number for n and keep doing the addition.

Answer 4

Use one of the half-angle formulas for tangent. Here's all of them: http://mathworld.wolfram.com/Half-AngleFormulas.html
This is just one of them: tan(1/2 x) = (1 - cos x)/sin x
Plugging in 30 degrees for x, you get:
tan((1/2)(30)) = (1 - cos 30)/sin 30
tan(15) = (1 - sqrt(3)/2)/(1/2) --- multiplying the top and bottom by 2 gives:
= (2 - sqrt(3))/1 = 2 - sqrt(3)
Hope this helps.

Answer 5

The way to solve this is to use the difference of two angles identity.

Tan (A-B)= (Tan A - Tan B)/(1+TanATanB)

15 degrees is the difference between 45 degrees and 30 degrees.

So Tan(15) = Tan(45-30) = (Tan(45) - Tan(30))/(1+Tan(45)Tan(30))

Tan(45) = 1
Tan (30)=1/2 / sqrt(3)/2 = sqrt(3)/3

Substitute:
Tan(15) = (1-sqrt(3)/3)
_________
(1+sqrt(3)/3)

Simplify:
= (3-sqrt(3))/3
__________
(3+sqrt(3))/3)

= 3-sqrt(3)
______
3+sqrt(3)


which is approximately equal to:

3-1.73
_____
3+ 1.73

=1.27
___
4.73

=.268


Hope this helps!

Answer 6

tan 15 = 0.26794919243112270647255365849413

:)

Answer 7

tan 15 = 0.267949192......

Answer 8

opposite over adjacent side or use tables

Answer 9

0.26794919243112270647255365849413
computed by calculator from computer

Answer 10

later you will see the differance trust me i no.