Respuesta :
Answer:
√3 - 2.
Step-by-step explanation:
Let A = 330 degrees so A/2 = 165 degrees.
tan A/2 = (1 - cos A) / sin A
tan 165 = (1 - cos 330) / sin 330
= (1 - √3/2) / (-1/2)
= -2(1 - √3/2)
= -2 + 2 * √3/2
= √3 - 2.
Answer:
[tex]\sqrt{3}[/tex] - 2
Step-by-step explanation:
Using the half- angle identity
tan( [tex]\frac{x}{2}[/tex] ) = [tex]\frac{sinx}{1+cosx}[/tex]
[tex]\frac{x}{2}[/tex] = 165° ⇒ x = 330°
sin330° = - sin30° = - [tex]\frac{1}{2}[/tex]
cos330° = cos30° = [tex]\frac{\sqrt{3} }{2}[/tex]
tan165° = [tex]\frac{sin330}{1+cos330}[/tex]
= [tex]\frac{-\frac{1}{2} }{1+\frac{\sqrt{3} }{2} }[/tex]
= - [tex]\frac{1}{2}[/tex] × [tex]\frac{2}{2+\sqrt{3} }[/tex]
= - [tex]\frac{1}{2+\sqrt{3} }[/tex]
Rationalise by multiplying numerator/ denominator by the conjugate of the denominator
The conjugate of 2 + [tex]\sqrt{3}[/tex] is 2 - [tex]\sqrt{3}[/tex], hence
tan 165°
= - [tex]\frac{2-\sqrt{3} }{(2+\sqrt{3})(2-\sqrt{3}) }[/tex]
= - [tex]\frac{2-\sqrt{3} }{4-3}[/tex]
= - (2 - [tex]\sqrt{3}[/tex] )
= - 2 + [tex]\sqrt{3}[/tex] = [tex]\sqrt{3}[/tex] - 2
