Use The Unit Circle To Find $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right$\] In Radians. Remember That The Domain Of Inverse Cosine Is Limited To Quadrants I And II (the Top Half Of The Unit Circle).A. $\frac{11 \pi}{6}$ B.

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Introduction

The unit circle is a fundamental concept in trigonometry, and it plays a crucial role in finding inverse trigonometric functions. In this article, we will explore how to use the unit circle to find the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) in radians. We will also discuss the domain of inverse cosine and how it relates to the unit circle.

Understanding the Domain of Inverse Cosine

The domain of inverse cosine is limited to quadrants I and II, which is the top half of the unit circle. This means that the value of cos⁡−1x\cos^{-1}x is only defined for xx values between −1-1 and 11, and the angle corresponding to the value of xx must lie in quadrants I and II.

Using the Unit Circle to Find Inverse Cosine Values

To find the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right), we need to find the angle corresponding to the value of −32-\frac{\sqrt{3}}{2} on the unit circle. We can do this by using the unit circle and the properties of cosine.

Finding the Angle Corresponding to the Value of −32-\frac{\sqrt{3}}{2}

The value of −32-\frac{\sqrt{3}}{2} corresponds to the angle 5π6\frac{5\pi}{6} on the unit circle. However, this angle lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.

Finding the Angle in Quadrant I

To find the angle in quadrant I that has the same value of cosine, we can use the fact that the cosine function is periodic with a period of 2π2\pi. This means that the angle in quadrant I that has the same value of cosine as 5π6\frac{5\pi}{6} is 5π6−2π=5π6−12π6=−7π6\frac{5\pi}{6} - 2\pi = \frac{5\pi}{6} - \frac{12\pi}{6} = -\frac{7\pi}{6}.

Finding the Angle in Quadrant II

However, the angle −7π6-\frac{7\pi}{6} lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant II that has the same value of cosine.

Finding the Angle in Quadrant II

To find the angle in quadrant II that has the same value of cosine, we can use the fact that the cosine function is an even function. This means that the angle in quadrant II that has the same value of cosine as −7π6-\frac{7\pi}{6} is π−7π6=3π6−7π6=−4π6=−2π3\pi - \frac{7\pi}{6} = \frac{3\pi}{6} - \frac{7\pi}{6} = -\frac{4\pi}{6} = -\frac{2\pi}{3}.

Finding the Angle in Quadrant I

However, the angle −2π3-\frac{2\pi}{3} lies in quadrant II, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.

Finding the Angle in Quadrant I

To find the angle in quadrant I that has the same value of cosine, we can use the fact that the cosine function is periodic with a period of 2π2\pi. This means that the angle in quadrant I that has the same value of cosine as −2π3-\frac{2\pi}{3} is 2π−2π3=6π3−2π3=4π32\pi - \frac{2\pi}{3} = \frac{6\pi}{3} - \frac{2\pi}{3} = \frac{4\pi}{3}.

Finding the Angle in Quadrant II

However, the angle 4π3\frac{4\pi}{3} lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant II that has the same value of cosine.

Finding the Angle in Quadrant II

To find the angle in quadrant II that has the same value of cosine, we can use the fact that the cosine function is an even function. This means that the angle in quadrant II that has the same value of cosine as 4π3\frac{4\pi}{3} is π−4π3=3π3−4π3=−π3\pi - \frac{4\pi}{3} = \frac{3\pi}{3} - \frac{4\pi}{3} = -\frac{\pi}{3}.

Finding the Angle in Quadrant I

However, the angle −π3-\frac{\pi}{3} lies in quadrant II, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.

Finding the Angle in Quadrant I

To find the angle in quadrant I that has the same value of cosine, we can use the fact that the cosine function is periodic with a period of 2π2\pi. This means that the angle in quadrant I that has the same value of cosine as −π3-\frac{\pi}{3} is 2π−π3=6π3−π3=5π32\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}.

Finding the Angle in Quadrant II

However, the angle 5π3\frac{5\pi}{3} lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant II that has the same value of cosine.

Finding the Angle in Quadrant II

To find the angle in quadrant II that has the same value of cosine, we can use the fact that the cosine function is an even function. This means that the angle in quadrant II that has the same value of cosine as 5π3\frac{5\pi}{3} is π−5π3=3π3−5π3=−2π3\pi - \frac{5\pi}{3} = \frac{3\pi}{3} - \frac{5\pi}{3} = -\frac{2\pi}{3}.

Finding the Angle in Quadrant I

However, the angle −2π3-\frac{2\pi}{3} lies in quadrant II, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.

Finding the Angle in Quadrant I

To find the angle in quadrant I that has the same value of cosine, we can use the fact that the cosine function is periodic with a period of 2π2\pi. This means that the angle in quadrant I that has the same value of cosine as −2π3-\frac{2\pi}{3} is 2π−2π3=6π3−2π3=4π32\pi - \frac{2\pi}{3} = \frac{6\pi}{3} - \frac{2\pi}{3} = \frac{4\pi}{3}.

Finding the Angle in Quadrant II

However, the angle 4π3\frac{4\pi}{3} lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant II that has the same value of cosine.

Finding the Angle in Quadrant II

To find the angle in quadrant II that has the same value of cosine, we can use the fact that the cosine function is an even function. This means that the angle in quadrant II that has the same value of cosine as 4π3\frac{4\pi}{3} is π−4π3=3π3−4π3=−π3\pi - \frac{4\pi}{3} = \frac{3\pi}{3} - \frac{4\pi}{3} = -\frac{\pi}{3}.

Finding the Angle in Quadrant I

However, the angle −π3-\frac{\pi}{3} lies in quadrant II, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.

Finding the Angle in Quadrant I

To find the angle in quadrant I that has the same value of cosine, we can use the fact that the cosine function is periodic with a period of 2π2\pi. This means that the angle in quadrant I that has the same value of cosine as −π3-\frac{\pi}{3} is 2π−π3=6π3−π3=5π32\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}.

Finding the Angle in Quadrant II

However, the angle 5π3\frac{5\pi}{3} lies in quadrant III, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant II that has the same value of cosine.

Finding the Angle in Quadrant II

To find the angle in quadrant II that has the same value of cosine, we can use the fact that the cosine function is an even function. This means that the angle in quadrant II that has the same value of cosine as 5π3\frac{5\pi}{3} is π−5π3=3π3−5π3=−2π3\pi - \frac{5\pi}{3} = \frac{3\pi}{3} - \frac{5\pi}{3} = -\frac{2\pi}{3}.

Finding the Angle in Quadrant I

However, the angle −2π3-\frac{2\pi}{3} lies in quadrant II, which is outside the domain of inverse cosine. Therefore, we need to find the angle in quadrant I that has the same value of cosine.


# **Using the Unit Circle to Find Inverse Cosine Values: Q&A**

Introduction

In our previous article, we explored how to use the unit circle to find the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) in radians. We also discussed the domain of inverse cosine and how it relates to the unit circle. In this article, we will answer some common questions related to using the unit circle to find inverse cosine values.

Q: What is the domain of inverse cosine?

A: The domain of inverse cosine is limited to quadrants I and II, which is the top half of the unit circle. This means that the value of cos⁡−1x\cos^{-1}x is only defined for xx values between −1-1 and 11, and the angle corresponding to the value of xx must lie in quadrants I and II.

Q: How do I find the angle corresponding to a given value of cosine on the unit circle?

A: To find the angle corresponding to a given value of cosine on the unit circle, you can use the unit circle and the properties of cosine. You can also use a calculator or a trigonometric table to find the angle.

Q: What if the angle corresponding to a given value of cosine lies in quadrant III or IV?

A: If the angle corresponding to a given value of cosine lies in quadrant III or IV, you need to find the angle in quadrant I or II that has the same value of cosine. You can do this by using the fact that the cosine function is periodic with a period of 2π2\pi.

Q: How do I use the fact that the cosine function is periodic with a period of 2π2\pi to find the angle in quadrant I or II that has the same value of cosine?

A: To use the fact that the cosine function is periodic with a period of 2π2\pi to find the angle in quadrant I or II that has the same value of cosine, you can subtract or add multiples of 2π2\pi to the angle corresponding to the given value of cosine.

Q: What if I get a negative angle in quadrant I or II?

A: If you get a negative angle in quadrant I or II, you can add 2π2\pi to the angle to get a positive angle in the same quadrant.

Q: Can I use the unit circle to find inverse cosine values for any value of xx between −1-1 and 11?

A: Yes, you can use the unit circle to find inverse cosine values for any value of xx between −1-1 and 11. However, you need to make sure that the angle corresponding to the value of xx lies in quadrants I and II.

Q: How do I know if the angle corresponding to a given value of cosine lies in quadrants I and II?

A: To know if the angle corresponding to a given value of cosine lies in quadrants I and II, you can use the unit circle and the properties of cosine. You can also use a calculator or a trigonometric table to find the angle.

Q: Can I use the unit circle to find inverse cosine values for values of xx outside the range −1-1 to 11?

A: No, you cannot use the unit circle to find inverse cosine values for values of xx outside the range −1-1 to 11. The domain of inverse cosine is limited to quadrants I and II, which is the top half of the unit circle.

Q: What if I get a value of xx outside the range −1-1 to 11 when using the unit circle to find inverse cosine values?

A: If you get a value of xx outside the range −1-1 to 11 when using the unit circle to find inverse cosine values, you need to check if the angle corresponding to the value of xx lies in quadrants I and II. If it does not, you need to find the angle in quadrant I or II that has the same value of cosine.

Conclusion

In this article, we answered some common questions related to using the unit circle to find inverse cosine values. We discussed the domain of inverse cosine, how to find the angle corresponding to a given value of cosine on the unit circle, and how to use the fact that the cosine function is periodic with a period of 2π2\pi to find the angle in quadrant I or II that has the same value of cosine. We also discussed what to do if you get a negative angle in quadrant I or II, and how to know if the angle corresponding to a given value of cosine lies in quadrants I and II.