What is the value of tan(arccos(-(sqrt(3))/2)+pi) ?

The value of tan(arccos(-(sqrt(3))/2)+pi) is -(sqrt(3))/3

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tan(arccos(-(sqrt(3))/2)+pi)

tan (arccos (- \frac{3}{2}) + π)

- \frac{3}{3}

Decimal Notation

- 0.57735 \dots

Solution steps

tan (arccos (- \frac{3}{2}) + π)

Rewrite using trig identities:

\frac{tan ( arccos ( - \frac{3}{2} ) ) + tan ( 0 )}{1 - tan ( arccos ( - \frac{3}{2} ) ) tan ( 0 )}

tan (arccos (- \frac{3}{2}) + π)

Use the Angle Sum identity:

tan (s + t) = \frac{tan ( s ) + tan ( t )}{1 - tan ( s ) tan ( t )}

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

tan (π) = tan (0)

tan (π)

Rewrite

π

π + 0

= tan (π + 0)

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + 0) = tan (0)

= tan (0)

= \frac{tan ( arccos ( - \frac{3}{2} ) ) + tan ( 0 )}{1 - tan ( arccos ( - \frac{3}{2} ) ) tan ( π )}

tan (π) = tan (0)

tan (π)

Rewrite

π

π + 0

= tan (π + 0)

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + 0) = tan (0)

= tan (0)

= \frac{tan ( arccos ( - \frac{3}{2} ) ) + tan ( 0 )}{1 - tan ( arccos ( - \frac{3}{2} ) ) tan ( 0 )}

= \frac{tan ( arccos ( - \frac{3}{2} ) ) + tan ( 0 )}{1 - tan ( arccos ( - \frac{3}{2} ) ) tan ( 0 )}

Rewrite using trig identities:

tan (arccos (- \frac{3}{2})) = - \frac{3}{3}

tan (arccos (- \frac{3}{2}))

Rewrite using trig identities:

tan (arccos (- \frac{3}{2})) = \frac{1 - ( - \frac{3}{2} ) ^{2}}{( - \frac{3}{2} )}

Use the following identity:

tan (arccos (x)) = \frac{1 - x ^{2}}{x}

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

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

Simplify

= - \frac{3}{3}

Use the following trivial identity:

tan (0) = 0

tan (0)

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= \frac{- \frac{3}{3} + 0}{1 - ( - \frac{3}{3} ) \cdot 0}

Simplify

= - \frac{3}{3}

3 + 2 sin (\frac{π}{3})

cosh (0) - 1

2 sec^{2} (\frac{π}{2}) tan (\frac{π}{2})

\frac{1 - sin ( 3 0 ^{\circ} )}{1 + sin ( 3 0 ^{\circ} )}

2 arctan (\frac{1}{2})

What is the value of tan(arccos(-(sqrt(3))/2)+pi) ?
The value of tan(arccos(-(sqrt(3))/2)+pi) is -(sqrt(3))/3