Problem

How would you use the unit circle to find $\sin \left(\frac{5 \pi}{6}\right)$ ?

Answer

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Answer

Final Answer: \(\boxed{0.5}\)

Steps

Step 1 :The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. The x-coordinate of a point on the unit circle represents the cosine of the angle formed by the positive x-axis and the line segment connecting the origin and the point, and the y-coordinate represents the sine of the angle.

Step 2 :To find \(\sin \left(\frac{5 \pi}{6}\right)\), we need to find the y-coordinate of the point on the unit circle that is \(\frac{5 \pi}{6}\) radians from the positive x-axis.

Step 3 :Since \(\frac{5 \pi}{6}\) is in the second quadrant where sine values are positive, and it is \(\frac{\pi}{6}\) radians away from \(\pi\), which corresponds to the point (-1, 0) on the unit circle, we can use the symmetry of the unit circle to find that \(\sin \left(\frac{5 \pi}{6}\right)\) is the same as \(\sin \left(\frac{\pi}{6}\right)\).

Step 4 :The result from the calculation is approximately 0.5, which is the expected value of \(\sin \left(\frac{\pi}{6}\right)\). This means that \(\sin \left(\frac{5 \pi}{6}\right)\) is also approximately 0.5.

Step 5 :Final Answer: \(\boxed{0.5}\)

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