Find The Exact Value Of \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right ]. Sin ⁡ − 1 ( − 3 2 ) = □ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \square Sin − 1 ( − 2 3 ​ ​ ) = □ (Type Your Answer In Radians.)

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Introduction

In this article, we will delve into the world of inverse trigonometric functions and explore the exact value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right). Inverse trigonometric functions are used to find the angle whose trigonometric function is a given value. In this case, we are looking for the angle whose sine is −32-\frac{\sqrt{3}}{2}.

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are denoted by the prefix "arc" or "inverse". For example, the inverse sine function is denoted by sin⁡−1x\sin^{-1}x. The inverse sine function returns the angle whose sine is equal to the given value. In other words, if sin⁡θ=x\sin\theta = x, then sin⁡−1x=θ\sin^{-1}x = \theta.

The Sine Function

The sine function is a periodic function that oscillates between −1-1 and 11. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function has a period of 2π2\pi, which means that the sine function repeats itself every 2π2\pi radians.

The Inverse Sine Function

The inverse sine function is defined as the angle whose sine is equal to the given value. The inverse sine function is denoted by sin⁡−1x\sin^{-1}x. The range of the inverse sine function is [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], which means that the inverse sine function returns an angle between −π2-\frac{\pi}{2} and π2\frac{\pi}{2} radians.

Finding the Exact Value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)

To find the exact value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right), we need to find the angle whose sine is equal to −32-\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 11 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle inscribed in the unit circle.

Finding the Angle

To find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}, we need to find the angle whose sine is equal to 32\frac{\sqrt{3}}{2} and then take the negative of that angle. We can use the unit circle to find this angle.

The Angle Whose Sine is Equal to 32\frac{\sqrt{3}}{2}

The angle whose sine is equal to 32\frac{\sqrt{3}}{2} is π3\frac{\pi}{3} radians. This is because the sine of π3\frac{\pi}{3} radians is equal to 32\frac{\sqrt{3}}{2}.

The Angle Whose Sine is Equal to −32-\frac{\sqrt{3}}{2}

Since the sine function is an odd function, we can take the negative of the angle whose sine is equal to 32\frac{\sqrt{3}}{2} to find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}. Therefore, the angle whose sine is equal to −32-\frac{\sqrt{3}}{2} is −π3-\frac{\pi}{3} radians.

Conclusion

In this article, we found the exact value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) to be −π3-\frac{\pi}{3} radians. We used the unit circle to find this angle and took the negative of the angle whose sine is equal to 32\frac{\sqrt{3}}{2} to find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}.

Final Answer

The final answer is: −π3\boxed{-\frac{\pi}{3}}

Q: What is the inverse sine function?

A: The inverse sine function, denoted by sin⁡−1x\sin^{-1}x, is a function that returns the angle whose sine is equal to the given value. In other words, if sin⁡θ=x\sin\theta = x, then sin⁡−1x=θ\sin^{-1}x = \theta.

Q: What is the range of the inverse sine function?

A: The range of the inverse sine function is [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], which means that the inverse sine function returns an angle between −π2-\frac{\pi}{2} and π2\frac{\pi}{2} radians.

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

A: To find the exact value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right), you need to find the angle whose sine is equal to −32-\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 11 centered at the origin of the coordinate plane. The unit circle is used to define the trigonometric functions.

Q: How do I use the unit circle to find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}?

A: To find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}, you need to find the angle whose sine is equal to 32\frac{\sqrt{3}}{2} and then take the negative of that angle. You can use the unit circle to find this angle.

Q: What is the angle whose sine is equal to 32\frac{\sqrt{3}}{2}?

A: The angle whose sine is equal to 32\frac{\sqrt{3}}{2} is π3\frac{\pi}{3} radians.

Q: What is the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}?

A: Since the sine function is an odd function, you can take the negative of the angle whose sine is equal to 32\frac{\sqrt{3}}{2} to find the angle whose sine is equal to −32-\frac{\sqrt{3}}{2}. Therefore, the angle whose sine is equal to −32-\frac{\sqrt{3}}{2} is −π3-\frac{\pi}{3} radians.

Q: What is the final answer to sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)?

A: The final answer to sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) is −π3-\frac{\pi}{3} radians.

Q: Can I use a calculator to find the value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)?

A: Yes, you can use a calculator to find the value of sin⁡−1(−32)\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right). However, keep in mind that the calculator may return an approximate value, whereas the exact value is −π3-\frac{\pi}{3} radians.

Q: What is the significance of the inverse sine function in real-world applications?

A: The inverse sine function has many real-world applications, such as calculating the angle of elevation of a building, the angle of depression of a projectile, and the angle of incidence of a wave.

Q: Can I use the inverse sine function to solve problems involving right triangles?

A: Yes, you can use the inverse sine function to solve problems involving right triangles. The inverse sine function can be used to find the angle of a right triangle given the length of the opposite side and the hypotenuse.

Q: Can I use the inverse sine function to solve problems involving periodic functions?

A: Yes, you can use the inverse sine function to solve problems involving periodic functions. The inverse sine function can be used to find the angle of a periodic function given the value of the function at a specific point.

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

A: Some common mistakes to avoid when using the inverse sine function include:

  • Not checking the range of the inverse sine function
  • Not using the correct quadrant for the angle
  • Not taking the negative of the angle when the sine function is negative
  • Not using the unit circle to find the angle

Q: How can I practice using the inverse sine function?

A: You can practice using the inverse sine function by solving problems involving right triangles, periodic functions, and other applications. You can also use online resources, such as calculators and graphing software, to visualize and explore the inverse sine function.