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

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

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Answer 1 · 2018-03-27T11:16:09

$tan (15^{\circ}) = \sqrt{3}$ $tan(15^circ)=sqrt(3)$

Explanation:

Half angle formulae for $\frac{tan (θ)}{2}$ $(tan(theta))/2$

$XXX = \frac{sin (θ)}{1 + cos (θ)} xxxxxxxx$ $color(white)("XXX")=(sin(theta))/(1+cos(theta))color(white)("xxxxxxxx")$ [1]

$XXX = \frac{cos (θ)}{1 - sin (θ)} xxxxxxxx$ $color(white)("XXX")=(cos(theta))/(1-sin(theta))color(white)("xxxxxxxx")$ [2]

$XXX = \pm \frac{\sqrt{1 - cos (θ)}}{1 + cos (θ)} xxxx$ $color(white)("XXX")=+-sqrt(1-cos(theta))/(1+cos(theta))color(white)("xxxx")$ [3]

Since $sin (30^{\circ}) = \frac{1}{2} xx$ $sin(30^circ)=1/2color(white)("xx")$ and $xx cos (30^{\circ}) = \frac{\sqrt{3}}{2}$ $color(white)("xx")cos(30^circ)=sqrt(3)/2$

We can use [2] (for example) to get
$tan (15^{\circ}) = tan (\frac{30^{\circ}}{2}) = \frac{(\frac{\sqrt{3}}{2})}{(1 - \frac{1}{2})} = \sqrt{3}$ $tan(15^circ)=tan((30^circ)/2)=((sqrt(3)/2))/((1-1/2))=sqrt(3)$

Answer 2 · 2018-03-28T05:13:35

tan 15 = (2 - sqrt3)

Explanation:

Use the half angle formula:
$tan (\frac{a}{2}) = \frac{1 - cos a}{sin a}$ $tan (a/2) = (1 - cos a)/sin a$
In this case --> $tan (\frac{a}{2}) = tan 15$ $tan (a/2) = tan 15$ --> $cos a = cos 30 = \frac{\sqrt{3}}{2}$ $cos a = cos 30 = sqrt3/2$
--> $sin a = sin 30 = \frac{1}{2}$ $sin a = sin 30 = 1/2$ .
The formula becomes:
$tan 15 = \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}} = 2 - \sqrt{3}$ $tan 15 = (1 - sqrt3/2)/(1/2) = 2 - sqrt3$
Check by calculator.
$2 - \sqrt{3} = 2 - 1.732 = 0.267$ $2 - sqrt3 = 2 - 1.732 = 0.267$
tan 15 = 0.267. Proved.

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

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