Find All Possible Values For Each Expression. Sin ⁡ − 1 1 2 \sin^{-1} \frac{1}{2} Sin − 1 2 1 ​ A. 30 ∘ + 180 N , 150 ∘ + 180 N 30^{\circ} + 180n, 150^{\circ} + 180n 3 0 ∘ + 180 N , 15 0 ∘ + 180 N B. 30 ∘ + 360 N , 150 ∘ + 360 N 30^{\circ} + 360n, 150^{\circ} + 360n 3 0 ∘ + 360 N , 15 0 ∘ + 360 N C. 60 ∘ + 180 N , 120 ∘ + 180 N 60^{\circ} + 180n, 120^{\circ} + 180n 6 0 ∘ + 180 N , 12 0 ∘ + 180 N D. $60^{\circ} +

Introduction

Trigonometric functions and their inverses are essential concepts in mathematics, particularly in trigonometry. In this article, we will focus on finding all possible values for each expression involving the inverse sine function, denoted as $\sin^{-1} x$. We will explore the different options and determine the correct range of values for each expression.

Understanding the Inverse Sine Function

The inverse sine function, denoted as $\sin^{-1} x$, is the inverse of the sine function. It returns the angle whose sine is a given value. The range of the inverse sine function is typically restricted to the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ to ensure a unique output for each input.

Option A: $30^{\circ} + 180n, 150^{\circ} + 180n$

To determine the possible values for the expression $\sin^{-1} \frac{1}{2}$, we need to find the angles whose sine is equal to $\frac{1}{2}$. We know that the sine of $30^{\circ}$ is equal to $\frac{1}{2}$. Therefore, one possible value for the expression is $30^{\circ}$. However, we also need to consider the periodic nature of the sine function.

The sine function has a period of $360^{\circ}$, which means that the sine of an angle is equal to the sine of that angle plus or minus any multiple of $360^{\circ}$. Therefore, we can add or subtract multiples of $360^{\circ}$ to the angle $30^{\circ}$ to obtain other possible values.

However, we need to be careful when adding or subtracting multiples of $360^{\circ}$. If we add or subtract an even multiple of $360^{\circ}$, we will obtain an angle that is coterminal with the original angle. For example, if we add $360^{\circ}$ to $30^{\circ}$, we obtain $390^{\circ}$, which is coterminal with $30^{\circ}$.

On the other hand, if we add or subtract an odd multiple of $360^{\circ}$, we will obtain an angle that is not coterminal with the original angle. For example, if we add $180^{\circ}$ to $30^{\circ}$, we obtain $210^{\circ}$, which is not coterminal with $30^{\circ}$.

Therefore, the possible values for the expression $\sin^{-1} \frac{1}{2}$ are $30^{\circ} + 180n$ and $150^{\circ} + 180n$, where $n$ is an integer.

Option B: $30^{\circ} + 360n, 150^{\circ} + 360n$

We can also consider the possibility that the expression $\sin^{-1} \frac{1}{2}$ has a period of $360^{\circ}$, rather than $180^{\circ}$. In this case, we can add or subtract multiples of $360^{\circ}$ to the angle $30^{\circ}$ to obtain other possible values.

However, we need to be careful when adding or subtracting multiples of $360^{\circ}$. If we add or subtract an even multiple of $360^{\circ}$, we will obtain an angle that is coterminal with the original angle. For example, if we add $360^{\circ}$ to $30^{\circ}$, we obtain $390^{\circ}$, which is coterminal with $30^{\circ}$.

On the other hand, if we add or subtract an odd multiple of $360^{\circ}$, we will obtain an angle that is not coterminal with the original angle. For example, if we add $180^{\circ}$ to $30^{\circ}$, we obtain $210^{\circ}$, which is not coterminal with $30^{\circ}$.

Therefore, the possible values for the expression $\sin^{-1} \frac{1}{2}$ are $30^{\circ} + 360n$ and $150^{\circ} + 360n$, where $n$ is an integer.

Option C: $60^{\circ} + 180n, 120^{\circ} + 180n$

We can also consider the possibility that the expression $\sin^{-1} \frac{1}{2}$ has a period of $180^{\circ}$, rather than $360^{\circ}$. In this case, we can add or subtract multiples of $180^{\circ}$ to the angle $60^{\circ}$ to obtain other possible values.

However, we need to be careful when adding or subtracting multiples of $180^{\circ}$. If we add or subtract an even multiple of $180^{\circ}$, we will obtain an angle that is coterminal with the original angle. For example, if we add $180^{\circ}$ to $60^{\circ}$, we obtain $240^{\circ}$, which is coterminal with $60^{\circ}$.

On the other hand, if we add or subtract an odd multiple of $180^{\circ}$, we will obtain an angle that is not coterminal with the original angle. For example, if we add $180^{\circ}$ to $60^{\circ}$, we obtain $240^{\circ}$, which is not coterminal with $60^{\circ}$.

Therefore, the possible values for the expression $\sin^{-1} \frac{1}{2}$ are $60^{\circ} + 180n$ and $120^{\circ} + 180n$, where $n$ is an integer.

Conclusion

In conclusion, the possible values for the expression $\sin^{-1} \frac{1}{2}$ are $30^{\circ} + 180n$ and $150^{\circ} + 180n$, where $n$ is an integer. This is because the sine of $30^{\circ}$ is equal to $\frac{1}{2}$, and the sine function has a period of $360^{\circ}$.

Therefore, the correct answer is option A: $30^{\circ} + 180n, 150^{\circ} + 180n$.

Q: What is the inverse sine function?

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

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

A: The range of the inverse sine function is typically restricted to the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ to ensure a unique output for each input.

Q: How do I find the possible values for a trigonometric expression involving the inverse sine function?

A: To find the possible values for a trigonometric expression involving the inverse sine function, you need to find the angles whose sine is equal to the given value. You can use the unit circle or a trigonometric table to find the angles.

Q: What is the period of the sine function?

A: The period of the sine function is $360^{\circ}$ or $2\pi$ radians.

Q: How do I determine the possible values for an expression involving the inverse sine function when the period is $360^{\circ}$?

A: To determine the possible values for an expression involving the inverse sine function when the period is $360^{\circ}$, you can add or subtract multiples of $360^{\circ}$ to the angle whose sine is equal to the given value.

Q: How do I determine the possible values for an expression involving the inverse sine function when the period is $180^{\circ}$?

A: To determine the possible values for an expression involving the inverse sine function when the period is $180^{\circ}$, you can add or subtract multiples of $180^{\circ}$ to the angle whose sine is equal to the given value.

Q: What is the difference between coterminal angles and non-coterminal angles?

A: Coterminal angles are angles that have the same terminal side on the unit circle. Non-coterminal angles are angles that have different terminal sides on the unit circle.

Q: How do I determine whether an angle is coterminal or non-coterminal with another angle?

A: To determine whether an angle is coterminal or non-coterminal with another angle, you can add or subtract multiples of the period of the sine function to the angle. If the resulting angle has the same terminal side as the original angle, it is coterminal. If the resulting angle has a different terminal side, it is non-coterminal.

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

A: The range of the inverse sine function is significant because it ensures that the output of the inverse sine function is unique for each input. This is important because the inverse sine function is used to find the angle whose sine is equal to a given value.

Q: How do I use the inverse sine function to solve problems involving trigonometric expressions?

A: To use the inverse sine function to solve problems involving trigonometric expressions, you need to find the angle whose sine is equal to the given value. You can use the unit circle or a trigonometric table to find the angle.

Q: What are some common mistakes to avoid when working with trigonometric expressions involving the inverse sine function?

A: Some common mistakes to avoid when working with trigonometric expressions involving the inverse sine function include:

• Not considering the range of the inverse sine function
• Not using the correct period of the sine function
• Not determining whether an angle is coterminal or non-coterminal with another angle
• Not using the correct unit circle or trigonometric table to find the angle

Conclusion

In conclusion, finding possible values for trigonometric expressions involving the inverse sine function requires a thorough understanding of the inverse sine function, the range of the inverse sine function, and the period of the sine function. By following the steps outlined in this article, you can determine the possible values for a trigonometric expression involving the inverse sine function and avoid common mistakes.