What is the value of tan(345) ?

The value of tan(345) is -2+sqrt(3)

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tan(345)

tan (34 5^{\circ})

- 2 + 3

Solution steps

tan (34 5^{\circ})

tan (34 5^{\circ}) = tan (16 5^{\circ})

= tan (16 5^{\circ})

Rewrite using trig identities:

\frac{tan ( 10 5 ^{\circ} ) + tan ( 6 0 ^{\circ} )}{1 - tan ( 10 5 ^{\circ} ) tan ( 6 0 ^{\circ} )}

= \frac{tan ( 10 5 ^{\circ} ) + tan ( 6 0 ^{\circ} )}{1 - tan ( 10 5 ^{\circ} ) tan ( 6 0 ^{\circ} )}

Rewrite using trig identities:

tan (10 5^{\circ}) = - 2 - 3

Use the following trivial identity:

tan (6 0^{\circ}) = 3

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

Simplify

\frac{- 2 - 3 + 3}{1 - ( - 2 - 3 ) 3} : - 2 + 3

= - 2 + 3

2 sin (\frac{π}{2})

arccos (\frac{3}{4})

\frac{2 \cdot ( cos ( 4 5 ^{\circ} ) ) ^{2} + 3 \cdot csc ( 3 7 ^{\circ} )}{2 \cdot tan ( 4 5 ^{\circ} )}

cos (5 5^{\circ})

sin (- 12 0^{\circ})