What is the value of sin(-420) ?

The value of sin(-420) is -(sqrt(3))/2

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sin(-420)

sin (- 42 0^{\circ})

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

sin (- 42 0^{\circ})

Use the following property:

sin (- x) = - sin (x)

sin (- 42 0^{\circ}) = - sin (42 0^{\circ})

= - sin (42 0^{\circ})

sin (42 0^{\circ}) = sin (6 0^{\circ})

sin (42 0^{\circ})

Rewrite

42 0^{\circ}

36 0^{\circ} + 6 0^{\circ}

= sin (36 0^{\circ} + 6 0^{\circ})

Apply the periodicity of

sin

sin (x + 36 0^{\circ}) = sin (x)

sin (36 0^{\circ} + 6 0^{\circ}) = sin (6 0^{\circ})

= sin (6 0^{\circ})

= - sin (6 0^{\circ})

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= - \frac{3}{2}

cos (2) (\frac{π}{3})

arccos (\frac{3 π}{4})

tan (9 5^{\circ})

sin (arcsin (\frac{9}{41}) - arccos (- \frac{3}{5}))

\frac{1 + cos ( 4 5 ^{\circ} )}{2}