What is the exact value of sin 120 degrees?
Understand the Problem
The question is asking for the exact value of the sine of 120 degrees. To solve it, we can use the unit circle or the sine function properties.
Answer
The sine of 120 degrees is $\frac{\sqrt{3}}{2}$.
Answer for screen readers
The exact value of the sine of 120 degrees is $\frac{\sqrt{3}}{2}$.
Steps to Solve
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Identify the reference angle
The angle of 120 degrees is located in the second quadrant of the unit circle. The reference angle, which is the angle formed with the x-axis, can be found by subtracting 120 degrees from 180 degrees.
$$ 180^\circ - 120^\circ = 60^\circ $$
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Determine the sine value from the reference angle
The sine of an angle in the second quadrant is positive. Therefore, we will use the sine of the reference angle, which is 60 degrees.
The sine value of 60 degrees is given by:
$$ \sin(60^\circ) = \frac{\sqrt{3}}{2} $$
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Apply the sine sign rule
Since 120 degrees is in the second quadrant, the sine value is positive. Thus, we conclude that:
$$ \sin(120^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} $$
The exact value of the sine of 120 degrees is $\frac{\sqrt{3}}{2}$.
More Information
The sine function shows periodic behavior, meaning that angles that are evenly spaced in relation to multiples of 180 degrees (like 120 degrees) have predictable sine values. This is a useful property when working with the unit circle.
Tips
- Mistakenly assuming that the sine of an angle in the second quadrant is negative. Remember that sine is the y-coordinate on the unit circle, which is positive in the second quadrant.
- Forgetting to find the reference angle for calculations when dealing with angles greater than 90 degrees.
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