What is the value of sin(-5*pi) ?

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sin(-5*pi)

sin (- 5 \cdot π)

0

Solution steps

sin (- 5 π)

Use the following property:

sin (- x) = - sin (x)

sin (- 5 π) = - sin (5 π)

= - sin (5 π)

sin (5 π) = sin (π)

sin (5 π)

Rewrite

5 π

2 π \cdot 2 + π

= sin (2 π 2 + π)

Apply the periodicity of

sin

sin (x + 2 π \cdot k) = sin (x)

sin (2 π \cdot 2 + π) = sin (π)

= sin (π)

= - 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}

= 0

= - 0

Simplify

= 0

cot (3 8^{\circ})

8 \cdot cos (6 0^{\circ})

\frac{sin ( 6 4 ^{\circ} )}{8} + 48 ln (2) cos (6 4^{\circ})

sin (18. 4^{\circ})

5 cos (5 π)