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. 11 Π 6 \frac{11\pi}{6} 6 11 Π ​ B.

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Understanding the Domain of Inverse Cosine

The inverse cosine function, denoted as cos⁡−1(x)\cos^{-1}(x), is used to find the angle whose cosine is a given value. However, 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 angle we are looking for must lie between 00 and π\pi radians.

The Unit Circle and Inverse Cosine

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry and is used to define the values of sine, cosine, and tangent for various angles. By using the unit circle, we can find the values of inverse trigonometric functions, including inverse cosine.

Finding cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)

To find cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right), we need to find the angle whose cosine is −32-\frac{\sqrt{3}}{2}. We can use the unit circle to find this angle.

Using the Unit Circle to Find the Angle

The unit circle has a radius of 1 and is centered at the origin. The cosine of an angle is equal to the x-coordinate of the point on the unit circle corresponding to that angle. Since we are looking for the angle whose cosine is −32-\frac{\sqrt{3}}{2}, we need to find the point on the unit circle with an x-coordinate of −32-\frac{\sqrt{3}}{2}.

Finding the Point on the Unit Circle

The point on the unit circle with an x-coordinate of −32-\frac{\sqrt{3}}{2} is located in quadrant IV, which is the bottom right quadrant of the unit circle. However, since the domain of inverse cosine is limited to quadrants I and II, we need to find the angle in quadrant I that has the same cosine value.

Finding the Angle in Quadrant I

The angle in quadrant I that has the same cosine value as the point in quadrant IV is 5π6\frac{5\pi}{6}. However, this is not the correct answer, as the cosine of 5π6\frac{5\pi}{6} is 32\frac{\sqrt{3}}{2}, not −32-\frac{\sqrt{3}}{2}.

Finding the Correct Angle

To find the correct angle, we need to find the angle in quadrant II that has the same cosine value as the point in quadrant IV. The angle in quadrant II that has the same cosine value as the point in quadrant IV is 11π6\frac{11\pi}{6}.

Conclusion

In conclusion, the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) in radians is 11π6\frac{11\pi}{6}. This is because the angle in quadrant II that has the same cosine value as the point in quadrant IV is 11π6\frac{11\pi}{6}.

Example Use Case

The inverse cosine function has many practical applications in mathematics and science. For example, it can be used to find the angle of a right triangle given the length of the adjacent side and the hypotenuse. It can also be used to find the angle of a triangle given the lengths of two sides.

Common Mistakes

One common mistake when finding inverse cosine values is to forget that the domain of inverse cosine is limited to quadrants I and II. This can lead to incorrect answers, as the angle we are looking for must lie between 00 and π\pi radians.

Tips and Tricks

To find inverse cosine values, it is helpful to use the unit circle and to remember that the domain of inverse cosine is limited to quadrants I and II. It is also helpful to use a calculator or a trigonometric table to find the values of cosine for various angles.

Conclusion

In conclusion, the inverse cosine function is a powerful tool in mathematics and science. By using the unit circle and remembering that the domain of inverse cosine is limited to quadrants I and II, we can find the values of inverse cosine for various angles.

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 angle we are looking for must lie between 00 and π\pi radians.

Q: How do I find the value of cos⁡−1(x)\cos^{-1}(x) using the unit circle?

A: To find the value of cos⁡−1(x)\cos^{-1}(x) using the unit circle, you need to find the angle whose cosine is xx. You can do this by finding the point on the unit circle with an x-coordinate of xx and then finding the angle corresponding to that point.

Q: What is the difference between cos⁡−1(x)\cos^{-1}(x) and cos⁡(x)\cos(x)?

A: The main difference between cos⁡−1(x)\cos^{-1}(x) and cos⁡(x)\cos(x) is that cos⁡(x)\cos(x) gives the cosine of an angle, while cos⁡−1(x)\cos^{-1}(x) gives the angle whose cosine is xx. In other words, cos⁡(x)\cos(x) is an output function, while cos⁡−1(x)\cos^{-1}(x) is an input function.

Q: Can I use a calculator to find the value of cos⁡−1(x)\cos^{-1}(x)?

A: Yes, you can use a calculator to find the value of cos⁡−1(x)\cos^{-1}(x). Most calculators have a built-in inverse cosine function that you can use to find the value of cos⁡−1(x)\cos^{-1}(x).

Q: How do I find the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)?

A: To find the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right), you need to find the angle whose cosine is −32-\frac{\sqrt{3}}{2}. You can do this by finding the point on the unit circle with an x-coordinate of −32-\frac{\sqrt{3}}{2} and then finding the angle corresponding to that point.

Q: What is the value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)?

A: The value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) is 11π6\frac{11\pi}{6}.

Q: Can I use the unit circle to find the value of cos⁡−1(x)\cos^{-1}(x) for any value of xx?

A: No, you cannot use the unit circle to find the value of cos⁡−1(x)\cos^{-1}(x) for any value of xx. The unit circle only works for values of xx between −1-1 and 11.

Q: How do I know if the value of cos⁡−1(x)\cos^{-1}(x) is in radians or degrees?

A: The value of cos⁡−1(x)\cos^{-1}(x) is always in radians. If you want to convert it to degrees, you can use the conversion factor 180∘π\frac{180^\circ}{\pi}.

Q: Can I use the inverse cosine function to find the angle of a right triangle?

A: Yes, you can use the inverse cosine function to find the angle of a right triangle. If you know the length of the adjacent side and the hypotenuse, you can use the inverse cosine function to find the angle.

Q: What are some common mistakes to avoid when using the inverse cosine function?

A: Some common mistakes to avoid when using the inverse cosine function include forgetting that the domain of inverse cosine is limited to quadrants I and II, and using the wrong value of xx.

Q: How do I choose between using the inverse cosine function and the inverse sine function?

A: You should choose between using the inverse cosine function and the inverse sine function based on the problem you are trying to solve. If you are looking for the angle whose cosine is xx, you should use the inverse cosine function. If you are looking for the angle whose sine is xx, you should use the inverse sine function.

Q: Can I use the inverse cosine function to find the value of cos⁡(x)\cos(x) for any value of xx?

A: No, you cannot use the inverse cosine function to find the value of cos⁡(x)\cos(x) for any value of xx. The inverse cosine function only works for values of xx between −1-1 and 11.