91Ó°ÊÓ

To solve problems involving angles, the first step is identifying their quadrant. Angles can be represented in the coordinate plane using the standard position, which means the angle's vertex is at the origin, and its initial side lies along the positive x-axis.

In our example, we have \(\frac{2\pi}{3}\) radians. It's essential to recognize that this is more than \(\pi/2\) (the angle for the positive y-axis) and less than \(\pi\) (the angle for the negative x-axis). Thus, \(\frac{2\pi}{3}\) is located in the second quadrant.

The quadrant helps determine the sign of trigonometric functions. For instance:

• In the first quadrant, all functions are positive.
• In the second quadrant, sine and cosecant are positive.
• In the third quadrant, tangent and cotangent are positive.
• In the fourth quadrant, cosine and secant are positive.

Identifying the correct quadrant is crucial as it will guide us to apply the correct sign to our final answer.

Radian measure is a method of measuring angles based on the length of the arc created by that angle on a unit circle. One full circle is \(2\pi\) radians, equivalent to \(360^\circ\). Understanding radian measures is key to solving trigonometric problems because it provides a more natural foundation for math and physics when compared to degrees.

For the angle \(\frac{2\pi}{3}\), this measurement tells us how far along the unit circle the angle extends. It's a way to visualize and compute angles without converting to degrees. This angle stops two-thirds of the way to \(\pi\) radians, which corresponds to half a circle (i.e., \(180^\circ\)). This intuitive approach helps when performing calculations, especially in calculus.

When working with radians, remember that:

• \(\pi/6 = 30^\circ\)
• \(\pi/4 = 45^\circ\)
• \(\pi/3 = 60^\circ\)
• \(\pi/2 = 90^\circ\)

Mastering these conversions can greatly assist in quickly identifying reference angles and locating them on the circle.

Reference angles are all about finding a comparable angle that lies in the first quadrant. This simplifies the calculation of trigonometric functions while considering the sign dictated by the quadrant. To find the reference angle for an angle in the second quadrant, subtract the angle from \(\pi\).

In our exercise, the given angle is \(\frac{2\pi}{3}\). By subtracting from \(\pi\), the reference angle is computed as:

\[ \pi - \frac{2\pi}{3} = \frac{\pi}{3} \] This means, regardless of the original location or direction on the coordinate plane of the angle, we can use the well-known trigonometric values of \(\frac{\pi}{3}\) to solve the problem easily.

Once the reference angle is determined, we use:

• \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
• \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\)
• \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\)

Lastly, ensure you apply the correct sign for the trigonometric function based on the location of the original angle, which in the second quadrant means the sine remains positive.