Find the angle corresponding to the cosine value of $-frac{2}{3}$ on the unit circle.

To find the angle corresponding to the cosine value of $-\frac{2}{3}$ on the unit circle, we start with the definition of cosine in terms of the unit circle.

The cosine of an angle is the x-coordinate of the point on the unit circle. Therefore, we need to determine the angles whose x-coordinate is $-\frac{2}{3}$.

Since cosine is negative in the second and third quadrants, the angles we are looking for are in these quadrants.

1. First angle: Let θ be the angle in the second quadrant.

$\theta = \cos^{-1} \left( -\frac{2}{3} \right) $

Using a calculator, we find that

$\theta \approx 131.81 ^\circ $

2. Second angle: In the third quadrant, the reference angle is the same, but we add 180 degrees.

$\theta = 180 ^\circ + 48.19 ^\circ = 228.19 ^\circ$

Therefore, the two angles are approximately 131.81° and 228.19°.

To solve for the angle with a cosine of $-frac{2}{3}$, we utilize the unit circle properties.

Cosine represents the x-coordinate of a point on the unit circle. Thus, we seek angles where the x-coordinate is $-frac{2}{3}$.

These angles will be in the second and third quadrants.

1. For the second quadrant angle, θ:

$ heta = arccos left( -frac{2}{3}
ight) $

From a calculator, this evaluates to approximately:

$ heta approx 2.3 ext{ radians} $

2. For the third quadrant angle, we add π to the reference angle:

$ heta = pi + (pi – 2.3)$

Simplifying, we get:

$ heta approx 3.98 ext{ radians} $

Thus, the angles are approximately 2.3 radians and 3.98 radians.

To find the angle with cosine $-frac{2}{3}$, we use the unit circle.

The angles with this x-coordinate are in the second and third quadrants.

In the second quadrant:

$ heta = arccos left( -frac{2}{3}
ight) approx 2.3 ext{ radians} $

In the third quadrant:

$ heta = pi + (pi – 2.3) approx 3.98 ext{ radians} $

Hence, the angles are approximately 2.3 and 3.98 radians.

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