How do you find the exact value of sin(4pi/3) without rationalizing

Answer 1

Some students can visualize degrees better than radians, so
4π/3 radians = 240°
240° = 180° + 60°
and we know from the standard 30-60-90 triangle that sin 60° = √3/2
but 240° is in quadrant III, where the sine is negative

thus sin 240° = -√3/2
or
sin(4π/3) = -√3/2

Answer 2

6 π / 3 = 360º ... 4 π / 3 = 240º

a 60º angle in Quad III

sin(60º) = ? ... and, sine is negative in Quad III

Answer 3

4 π / 3 = π + π / 3

Apply the formula:

sin ( A + B ) = sin A cos B + cos A sin B

in this case:

sin ( π + π / 3 ) = sin π ∙ cos π / 3 + cos π ∙ sin π / 3

sin 4 π / 3 = 0 ∙ cos π / 3 + ( - 1 ) ∙ sin π / 3

sin 4 π / 3 = - sin π / 3

sin 4 π / 3 = - √3 / 2

This was in radians.

If it's easier for you in degrees:

4 π / 3 ∙ 180 / π = 240

4 π / 3 rad = 240°

240° = 180° + 60°

sin ( A + B ) = sin A cos B + cos A sin B

in this case:

sin ( 180° + 60° ) = sin 180° ∙ cos 60° + cos 180° ∙ sin 60°

sin 240° = 0 ∙ cos 60° + ( - 1 ) ∙ sin 60°

sin 240° = - sin 60°

sin 240° = - √3 / 2