Finding The Exact Value Of Arcsin(-√2 2) In Radians And Degrees

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In mathematics, trigonometric functions play a crucial role in various fields such as physics, engineering, and computer science. Among these functions, the inverse trigonometric functions, also known as arc functions, are essential for finding angles corresponding to specific trigonometric ratios. In this article, we will delve into the process of finding the exact value of the expression sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)} in both radians and degrees. This comprehensive guide aims to provide a clear understanding of the concepts involved and the steps necessary to solve this type of problem. Understanding inverse trigonometric functions is fundamental for students and professionals alike, as they appear frequently in mathematical calculations and real-world applications. Whether you are a student learning trigonometry or a professional needing to apply these concepts, this article will serve as a valuable resource.

Understanding Inverse Trigonometric Functions

To effectively find the exact value of an expression like sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)}, it's crucial to first understand the concept of inverse trigonometric functions. The inverse sine function, denoted as sin⁡−1(x){\sin^{-1}(x)} or arcsin(x), returns the angle whose sine is x. However, it's essential to remember that the sine function is periodic, meaning it repeats its values at regular intervals. Consequently, there are infinitely many angles that have the same sine value. To address this, we define the principal value range for sin⁡−1(x){\sin^{-1}(x)} to be [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]} in radians, or [−90∘,90∘]{[-90^\circ, 90^\circ]} in degrees. This restriction ensures that the inverse sine function has a unique output for each input within its domain. The domain of sin⁡−1(x){\sin^{-1}(x)} is [−1,1]{[-1, 1]}, as the sine function's range is [−1,1]{[-1, 1]}. When we encounter an expression like sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)}, we are essentially asking: "What angle, within the principal value range, has a sine of −22{-\frac{\sqrt{2}}{2}}?" This understanding is the foundation for solving the problem and finding the exact value in both radians and degrees. The ability to grasp the concept of principal values and the domain restrictions is vital for accurately working with inverse trigonometric functions.

Key Concepts and Definitions

  • Inverse Sine Function: The inverse sine function, denoted as sin⁡−1(x){\sin^{-1}(x)} or arcsin(x), gives the angle whose sine is x.
  • Principal Value Range: The principal value range for sin⁡−1(x){\sin^{-1}(x)} is [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]} in radians, or [−90∘,90∘]{[-90^\circ, 90^\circ]} in degrees.
  • Domain of sin⁡−1(x){\sin^{-1}(x)}: The domain of sin⁡−1(x){\sin^{-1}(x)} is [−1,1]{[-1, 1]}.
  • Periodicity of Sine Function: The sine function is periodic, meaning it repeats its values at regular intervals.

Understanding these concepts is crucial for accurately solving problems involving inverse trigonometric functions and ensuring that the solutions fall within the defined principal value range.

Step-by-Step Solution in Radians

To find the exact value of sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)} in radians, we need to identify an angle within the principal value range [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]} whose sine is −22{-\frac{\sqrt{2}}{2}}. This involves a systematic approach that combines knowledge of trigonometric values and the unit circle. First, we recognize that 22{\frac{\sqrt{2}}{2}} is a common value associated with angles that are multiples of π4{\frac{\pi}{4}} (45 degrees). Specifically, we know that sin⁡(π4)=22{\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}}. However, our expression involves the negative value, −22{-\frac{\sqrt{2}}{2}}, which indicates that the angle lies in either the third or fourth quadrant, where the sine function is negative. Given the principal value range [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]}, we focus on the fourth quadrant, as it contains negative angles within this range. The angle in the fourth quadrant whose sine is −22{-\frac{\sqrt{2}}{2}} is −π4{-\frac{\pi}{4}}. Therefore, sin⁡−1(−22)=−π4{\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}}. This step-by-step process ensures that we not only find an angle with the correct sine value but also that the angle falls within the defined principal value range, leading to an accurate and unique solution.

Detailed Steps

  1. Identify the reference angle: Recognize that sin⁡(π4)=22{\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}}.
  2. Determine the quadrant: Since the value is negative, the angle lies in the third or fourth quadrant.
  3. Consider the principal value range: The principal value range for sin⁡−1(x){\sin^{-1}(x)} is [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]}.
  4. Find the angle in the correct quadrant: The angle in the fourth quadrant with a sine of −22{-\frac{\sqrt{2}}{2}} is −π4{-\frac{\pi}{4}}.
  5. Write the final answer: sin⁡−1(−22)=−π4{\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}} radians.

By following these steps, you can confidently find the exact value of inverse sine expressions in radians, ensuring that your solution is both accurate and within the defined principal value range.

Converting Radians to Degrees

After finding the exact value in radians, the next step is to convert it to degrees. This conversion is crucial for those who are more comfortable working with degrees or when the problem specifically asks for the answer in degrees. The conversion between radians and degrees is based on the fundamental relationship that π{\pi} radians is equal to 180 degrees. This relationship forms the basis of our conversion formula: degrees = radians × 180π{\frac{180}{\pi}}. Applying this to our previous result, −π4{-\frac{\pi}{4}} radians, we substitute this value into the formula: degrees = −π4{-\frac{\pi}{4}} × 180π{\frac{180}{\pi}}. The π{\pi} terms cancel out, simplifying the expression to degrees = −14{-\frac{1}{4}} × 180. Performing the multiplication, we get degrees = -45. Therefore, −π4{-\frac{\pi}{4}} radians is equivalent to -45 degrees. This conversion process is straightforward and essential for expressing angles in different units, ensuring clarity and accuracy in mathematical and practical contexts. Understanding how to convert between radians and degrees is a fundamental skill in trigonometry and is widely used in various applications, from navigation to engineering.

Conversion Formula and Steps

  1. Conversion Formula: degrees = radians × 180π{\frac{180}{\pi}}
  2. Substitute the radian value: Substitute −π4{-\frac{\pi}{4}} into the formula.
  3. Simplify the expression: degrees = −π4{-\frac{\pi}{4}} × 180π{\frac{180}{\pi}}
  4. Cancel out π{\pi}: degrees = −14{-\frac{1}{4}} × 180
  5. Perform the multiplication: degrees = -45
  6. Write the final answer: −π4{-\frac{\pi}{4}} radians = -45 degrees

By following these steps, you can easily convert radians to degrees, ensuring your answers are accurate and in the desired format.

Final Answer in Degrees

Having converted the radian value to degrees, we can now state the final answer in degrees. We have determined that sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)} is equal to -45 degrees. This result aligns with our understanding of the unit circle and the principal value range for the inverse sine function. The angle -45 degrees lies within the principal value range of [−90∘,90∘]{[-90^\circ, 90^\circ]}, which confirms the validity of our solution. The negative sign indicates that the angle is measured clockwise from the positive x-axis, placing it in the fourth quadrant where the sine function is negative. This step solidifies our understanding of the relationship between radians and degrees and provides a clear, concise answer to the problem. The ability to express the solution in both radians and degrees demonstrates a comprehensive grasp of trigonometric concepts and their applications. In practical contexts, understanding angles in both units is essential for various fields, including engineering, physics, and computer graphics.

Summary of the Solution

  • Radian Value: sin⁡−1(−22)=−π4{\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) = -\frac{\pi}{4}} radians
  • Conversion to Degrees: Using the formula degrees = radians × 180π{\frac{180}{\pi}}, we converted −π4{-\frac{\pi}{4}} radians to degrees.
  • Final Answer in Degrees: sin⁡−1(−22)=−45{\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right) = -45} degrees

This final answer provides a clear and accurate solution to the problem, expressed in degrees, and reinforces the understanding of the steps involved in finding the exact value of inverse trigonometric expressions.

Common Mistakes to Avoid

When finding the exact value of inverse trigonometric expressions, it's crucial to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is neglecting the principal value range. For sin⁡−1(x){\sin^{-1}(x)}, the principal value range is [−π2,π2]{[-\frac{\pi}{2}, \frac{\pi}{2}]} in radians or [−90∘,90∘]{[-90^\circ, 90^\circ]} in degrees. Failing to ensure that the final answer falls within this range can result in an incorrect solution. For instance, an angle outside this range might have the same sine value but is not the principal value. Another common mistake is confusing the signs of trigonometric functions in different quadrants. The sine function is negative in the third and fourth quadrants, but only the fourth quadrant falls within the principal value range for sin⁡−1(x){\sin^{-1}(x)}. Misidentifying the correct quadrant can lead to an incorrect angle. Additionally, errors in converting between radians and degrees are common. It's essential to remember the conversion formula and apply it accurately to avoid mistakes. Another pitfall is not recognizing common trigonometric values. Knowing the sine, cosine, and tangent of special angles like 0, π6{\frac{\pi}{6}}, π4{\frac{\pi}{4}}, π3{\frac{\pi}{3}}, and π2{\frac{\pi}{2}} (or 0°, 30°, 45°, 60°, and 90°) is crucial for simplifying expressions and finding exact values. By being mindful of these common mistakes and practicing careful attention to detail, you can improve your accuracy and confidence in solving inverse trigonometric problems.

List of Common Mistakes

  1. Neglecting the Principal Value Range: Failing to ensure the final answer is within the correct range.
  2. Sign Errors: Misunderstanding the signs of trigonometric functions in different quadrants.
  3. Incorrect Conversion: Errors in converting between radians and degrees.
  4. Forgetting Common Values: Not recognizing the sine, cosine, and tangent of special angles.
  5. Misidentifying the Quadrant: Choosing the wrong quadrant for the angle.

By being aware of these pitfalls and taking steps to avoid them, you can significantly improve your accuracy in solving inverse trigonometric problems.

Practice Problems

To reinforce your understanding of finding the exact value of inverse trigonometric expressions, working through practice problems is essential. These exercises allow you to apply the concepts and techniques discussed in this article and identify areas where you may need further clarification. Here are a few practice problems to get you started:

  1. Find the exact value of cos⁡−1(−32){\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)} in radians and degrees.
  2. Evaluate tan⁡−1(−1){\tan^{-1}(-1)} in radians and degrees.
  3. Determine the exact value of sin⁡−1(1){\sin^{-1}(1)} in radians and degrees.
  4. Calculate cos⁡−1(0){\cos^{-1}(0)} in radians and degrees.
  5. Find the exact value of tan⁡−1(3){\tan^{-1}(\sqrt{3})} in radians and degrees.

For each problem, remember to consider the principal value range for the respective inverse trigonometric function. The principal value range for cos⁡−1(x){\cos^{-1}(x)} is [0,π]{[0, \pi]} in radians or [0∘,180∘]{[0^\circ, 180^\circ]} in degrees, and for tan⁡−1(x){\tan^{-1}(x)} it is (−π2,π2){(-\frac{\pi}{2}, \frac{\pi}{2})} in radians or (−90∘,90∘){(-90^\circ, 90^\circ)} in degrees. Work through each problem step-by-step, showing your calculations and reasoning. After completing the problems, check your answers against the solutions to ensure accuracy. If you encounter any difficulties, review the relevant sections of this article or consult additional resources. Consistent practice is key to mastering inverse trigonometric functions and their applications.

Tips for Solving Practice Problems

  • Review the Concepts: Ensure you understand the definitions and principal value ranges of inverse trigonometric functions.
  • Use the Unit Circle: Refer to the unit circle to visualize angles and their trigonometric values.
  • Show Your Work: Write down each step of your solution to avoid errors and track your progress.
  • Check Your Answers: Compare your solutions with the correct answers to identify mistakes and learn from them.
  • Seek Help When Needed: If you're struggling with a problem, don't hesitate to ask for assistance from teachers, classmates, or online resources.

Conclusion

In conclusion, finding the exact value of inverse trigonometric expressions like sin⁡−1(−22){\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)} involves a systematic approach that combines understanding the definitions, principal value ranges, and trigonometric relationships. We have demonstrated step-by-step how to determine the exact value in both radians and degrees, emphasizing the importance of considering the principal value range and converting between radians and degrees accurately. By following these methods and avoiding common mistakes, you can confidently solve a wide range of inverse trigonometric problems. The ability to work with inverse trigonometric functions is a valuable skill in mathematics and has numerous applications in fields such as physics, engineering, and computer science. Practice is key to mastering these concepts, so we encourage you to work through additional problems and seek help when needed. With a solid understanding of inverse trigonometric functions, you'll be well-equipped to tackle more advanced mathematical concepts and real-world applications.

Key Takeaways

  • Understand the definition and principal value range of sin⁡−1(x){\sin^{-1}(x)}.
  • Recognize common trigonometric values for special angles.
  • Follow a step-by-step approach to find the exact value in radians.
  • Convert radians to degrees using the appropriate formula.
  • Avoid common mistakes by paying attention to signs, quadrants, and the principal value range.
  • Practice solving problems to reinforce your understanding.

By mastering these concepts and techniques, you can confidently solve inverse trigonometric problems and apply them in various contexts.