sin of 5pi/6
Understand the Problem
The question is asking for the value of the sine function evaluated at the angle of 5π/6 radians. To solve this, we can use the unit circle or the sine function's properties.
Answer
$ \frac{1}{2} $
Answer for screen readers
The value of $ \sin\left(\frac{5\pi}{6}\right) $ is $ \frac{1}{2} $.
Steps to Solve
- Identify the angle in standard position
The angle given is $ \frac{5\pi}{6} $ radians. In terms of degrees, this is equivalent to $ 150^\circ $, since $ \frac{5\pi}{6} \times \frac{180}{\pi} = 150 $.
- Locate the angle on the unit circle
The angle $ 150^\circ $ (or $ \frac{5\pi}{6} $ radians) is located in the second quadrant of the unit circle.
- Find the reference angle
The reference angle for $ 150^\circ $ is calculated as follows: $$ 180^\circ - 150^\circ = 30^\circ $$ In radians, this reference angle is $$ \pi - \frac{5\pi}{6} = \frac{\pi}{6}. $$
- Determine the sine of the reference angle
The sine of the reference angle $ \frac{\pi}{6} $ is: $$ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. $$
- Consider the sign of sine in the second quadrant
In the second quadrant, sine is positive. Therefore, we have: $$ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. $$
The value of $ \sin\left(\frac{5\pi}{6}\right) $ is $ \frac{1}{2} $.
More Information
The sine function is periodic and defined for all real numbers. In addition, understanding the unit circle helps in evaluating sine values for any angle.
Tips
- Confusing the reference angle with the given angle. Always ensure you are using the correct reference.
- Forgetting to consider the sign of sine in different quadrants.
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