Question

How do you use half angle formula to find tan 15?

Answer 1

`tan(15^circ)=sqrt(3)`

Explanation:

Half angle formulae for `(tan(theta))/2`

`color(white)("XXX")=(sin(theta))/(1+cos(theta))color(white)("xxxxxxxx")`[1]

`color(white)("XXX")=(cos(theta))/(1-sin(theta))color(white)("xxxxxxxx")`[2]

`color(white)("XXX")=+-sqrt(1-cos(theta))/(1+cos(theta))color(white)("xxxx")`[3]

Since `sin(30^circ)=1/2color(white)("xx")`and`color(white)("xx")cos(30^circ)=sqrt(3)/2`

We can use [2] (for example) to get
`tan(15^circ)=tan((30^circ)/2)=((sqrt(3)/2))/((1-1/2))=sqrt(3)`

Answer 2

tan 15 = (2 - sqrt3)

Explanation:

Use the half angle formula:
`tan (a/2) = (1 - cos a)/sin a`
In this case --> `tan (a/2) = tan 15` --> `cos a = cos 30 = sqrt3/2`
--> `sin a = sin 30 = 1/2`.
The formula becomes:
`tan 15 = (1 - sqrt3/2)/(1/2) = 2 - sqrt3`
Check by calculator.
`2 - sqrt3 = 2 - 1.732 = 0.267`
tan 15 = 0.267. Proved.