Evaluate $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$. Enter Your Answer In Radians.

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Introduction

In mathematics, the inverse sine function, denoted as $\operatorname{Sin}^{-1} x$, is the inverse of the sine function. It returns the angle whose sine is a given number. In this article, we will evaluate the inverse sine of $\frac{\sqrt{3}}{2}$ and express the result in radians.

Understanding the Inverse Sine Function

The inverse sine function is a mathematical function that takes a real number as input and returns an angle in radians. The sine function, on the other hand, takes an angle as input and returns a real number between -1 and 1. The inverse sine function is defined as the angle whose sine is equal to the input value.

Evaluating $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$

To evaluate $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$, we need to find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$. We can use the unit circle to find this angle.

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions, including the sine function. The sine of an angle is equal to the y-coordinate of the point on the unit circle that corresponds to the angle.

Finding the Angle

To find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$, we need to find the point on the unit circle that corresponds to this sine value. We can do this by using the Pythagorean identity, which states that $\sin^2 \theta + \cos^2 \theta = 1$.

Using the Pythagorean Identity

Using the Pythagorean identity, we can write:

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

Simplifying this equation, we get:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

Finding the Angle

Since the sine of the angle is positive, we know that the angle is in the first or second quadrant. We can use the unit circle to find the angle whose cosine is equal to $\frac{1}{2}$.

The Angle in the First Quadrant

The angle in the first quadrant whose cosine is equal to $\frac{1}{2}$ is $\frac{\pi}{3}$.

The Angle in the Second Quadrant

The angle in the second quadrant whose cosine is equal to $\frac{1}{2}$ is $\frac{2\pi}{3}$.

Evaluating $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$

Since the sine of the angle is positive, we know that the angle is in the first or second quadrant. We can use the unit circle to find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$.

The Angle in the First Quadrant

The angle in the first quadrant whose sine is equal to $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$.

The Angle in the Second Quadrant

The angle in the second quadrant whose sine is equal to $\frac{\sqrt{3}}{2}$ is $\frac{2\pi}{3}$.

Conclusion

In conclusion, the value of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$ or $\frac{2\pi}{3}$, depending on the quadrant in which the angle lies.

Final Answer

The final answer is $\boxed{\frac{\pi}{3}}$ or $\boxed{\frac{2\pi}{3}}$.

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Introduction

In our previous article, we evaluated the inverse sine of $\frac{\sqrt{3}}{2}$ and expressed the result in radians. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the inverse sine function?

A: The inverse sine function, denoted as $\operatorname{Sin}^{-1} x$, is the inverse of the sine function. It returns the angle whose sine is a given number.

Q: How do I evaluate $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$?

A: To evaluate $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$, you need to find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$. You can use the unit circle to find this angle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions, including the sine function.

Q: How do I find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$?

A: To find the angle whose sine is equal to $\frac{\sqrt{3}}{2}$, you can use the Pythagorean identity, which states that $\sin^2 \theta + \cos^2 \theta = 1$. You can then solve for $\theta$.

Q: What are the possible values of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$?

A: The possible values of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ are $\frac{\pi}{3}$ and $\frac{2\pi}{3}$, depending on the quadrant in which the angle lies.

Q: Why are there two possible values of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$?

A: There are two possible values of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ because the sine function is periodic, meaning that it repeats itself every $2\pi$ radians. Therefore, there are two angles in the first and second quadrants whose sine is equal to $\frac{\sqrt{3}}{2}$.

Q: How do I determine which value of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ is correct?

A: To determine which value of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ is correct, you need to consider the quadrant in which the angle lies. If the angle is in the first quadrant, the correct value is $\frac{\pi}{3}$. If the angle is in the second quadrant, the correct value is $\frac{2\pi}{3}$.

Q: What is the final answer to $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$?

A: The final answer to $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ is $\boxed{\frac{\pi}{3}}$ or $\boxed{\frac{2\pi}{3}}$, depending on the quadrant in which the angle lies.

Conclusion

In conclusion, evaluating $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ requires a thorough understanding of the inverse sine function and the unit circle. By following the steps outlined in this article, you can determine the correct value of $\operatorname{Sin}^{-1} \frac{\sqrt{3}}{2}$ and express the result in radians.