Answer
$ \tan 72^{\circ}=\frac{\sqrt{10+2\sqrt{5}} }{\sqrt{5}-1}=3.078 $
Work Step by Step
By using trigonometric ratios of complementary angles.
$ \cos \theta = \sin (90 ^{\circ} - \theta ) $
Let $ \theta =72^{\circ} $ .
$ \cos 72 ^{\circ} = \sin (90 ^{\circ} - 72 ) $
Add like terms.
$ \cos 72 ^{\circ} = \sin (18 ^{\circ}) $
Substitute the value of $ \sin (18 ^{\circ}) $ .
$ \cos 72 ^{\circ} = \frac{\sqrt{5}-1}{4} $
By using trigonometric basic identity.
$ \sin ^2 \theta + \cos ^2 \theta =1 $
Isolate $ \sin \theta $ .
$ \sin \theta =\sqrt{1- \cos ^2 \theta } $
Plug $ \theta = 72 ^{\circ} $ into the above equation.
$ \sin 72 ^{\circ} =\sqrt{1- \cos ^2 72 ^{\circ} } $
Substitute the value of $ \cos 72 ^{\circ} $.
$ \sin 72 ^{\circ} =\sqrt{1- \left (\frac{\sqrt{5}-1}{4} \right )^2 } $
$ \sin 72 ^{\circ} =\sqrt{1- \left (\frac{6-2\sqrt{5}}{16} \right ) } $
$ \sin 72 ^{\circ} =\sqrt{ \frac{10+2\sqrt{5}}{16} } $
$ \sin 72 ^{\circ} = \frac{\sqrt{10+2\sqrt{5}}}{4} $
By using trigonometric ratios.
$ \tan \theta= \frac{\sin \theta}{\cos \theta} $
Plug $ \theta = 72 ^{\circ} $ into the above equation.
$ \tan 72 ^{\circ}= \frac{\sin 72 ^{\circ}}{\cos 72 ^{\circ}} $
Substitute values.
$ \tan 72 ^{\circ} =\frac{\frac{\sqrt{10+2\sqrt{5}}}{4}}{\frac{\sqrt{5}-1}{4}} $
Simplify.
$ \tan 72 ^{\circ} =\frac{\sqrt{10+2\sqrt{5}}}{\sqrt{5}-1} $
Use calculator for exact value.
$ \tan 72 ^{\circ} =3.078 $
