What is the value of arcsin(sin(-pi)) ?

The value of arcsin(sin(-pi)) is 0

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arcsin(sin(-pi))

arcsin (sin (- π))

0

Solution steps

arcsin (sin (- π))

Use the following property:

sin (- x) = - sin (x)

sin (- π) = - sin (π)

= arcsin (- sin (π))

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- sin (π)) = - arcsin (sin (π))

= - arcsin (sin (π))

Rewrite using trig identities:

arcsin (sin (π)) = 0

arcsin (sin (π))

Use the following trivial identity:

sin (π) = 0

sin (π)

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= arcsin (0)

= arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

arcsin (0)

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

= 0

= - 0

Simplify

= 0

sec^{2} (arctan (2)) + csc^{2} (arctan (3))

cos (\frac{10 π}{9})

\frac{1}{2} (π - \frac{1}{2} sin (2 π))

cos (arcsin (\frac{1}{5}) + arctan (\frac{1}{3}))

sin (arcsin (\frac{5}{13}) + arccos (- \frac{3}{5}))