What is the value of tan(pi+arcsin(2/3)) ?

The value of tan(pi+arcsin(2/3)) is (2sqrt(5))/5

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tan(pi+arcsin(2/3))

tan (π + arcsin (\frac{2}{3}))

\frac{2 5}{5}

Decimal Notation

0.89442 \dots

Solution steps

tan (π + arcsin (\frac{2}{3}))

Rewrite using trig identities:

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

tan (π + arcsin (\frac{2}{3}))

Use the Angle Sum identity:

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

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

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

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

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

Rewrite using trig identities:

tan (arcsin (\frac{2}{3})) = \frac{2 5}{5}

tan (arcsin (\frac{2}{3}))

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 5}{5}

= \frac{0 + \frac{2 5}{5}}{1 - 0 \cdot \frac{2 5}{5}}

Simplify

= \frac{2 5}{5}

- sin^{2} (1)

20 tan (3 0^{\circ})

\frac{2}{tan ( 6 5 ^{\circ} )}

sin (\frac{7 π}{12}) cos (\frac{π}{12})

arccos (\frac{1}{58})

What is the value of tan(pi+arcsin(2/3)) ?
The value of tan(pi+arcsin(2/3)) is (2sqrt(5))/5