find the exact value of tan(14π/3) using the unit circle

Jaden Easton 2021-08-20 Answered

find the exact value of \(\tan(\frac{4\pi}{3}-\frac{\pi}{4})\)

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Expert Answer

Alix Ortiz
Answered 2021-08-21 Author has 9249 answers

Trigonometry:

\(\tan(\frac{4\pi}{3}-\frac{\pi}{4})\)

We know

\(\tan(a-b)=\frac{\tan a-\tan b}{1+\tan a\tan b}\)

\(\tan(\frac{4\pi}{3})=\tan(240^\circ)\)

\(=\tan(90\cdot2+60^\circ)\)

\(=\tan60^\circ\)

\(=\sqrt{3}\)

\(\tan(\frac{\pi}{4})=1\)

\(\tan(\frac{4\pi}{3}-\frac{\pi}{4})\)

\(=\frac{\tan\frac{4x}{3}-\tan\frac{x}{4}}{1+\tan\frac{4x}{3}\tan\frac{x}{4}}\)

\(=\frac{\sqrt{3}-1}{1+\sqrt{3}}\)

\(=\frac{(\sqrt{3}-1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}\)

\(=\frac{3-2\sqrt{3}+1}{3-1}\)

\(=\frac{4-2\sqrt{3}}{2}\)

\(=2-\sqrt{3}\)

\(=0.2679\)

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