How do you find the exact value of sin(4pi/3) without rationalizing
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
6 π / 3 = 360º ... 4 π / 3 = 240º
a 60º angle in Quad III
sin(60º) = ? ... and, sine is negative in Quad III
To find the exact value of sin(4π/3) without rationalizing, we can use the unit circle and trigonometric identities.
1. Start by drawing a unit circle, which is a circle with a radius of 1 centered at the origin (0, 0) on a Cartesian coordinate system.
2. Locate the angle 4π/3 on the unit circle. This angle is measured counterclockwise from the positive x-axis.
3. To determine the value of sin(4π/3), we need to find the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
4. In the case of 4π/3, the terminal side intersects the unit circle at the point (-√3/2, -1/2), as indicated by the x and y coordinates.
5. The y-coordinate of the point gives us the sine value of the angle. Therefore, sin(4π/3) = -1/2.
So, the exact value of sin(4π/3) without rationalizing is -1/2.
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