Question 1 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: sin⁻¹(-1/2)

Correct Answer

-π6\frac{\pi}{6}

Exercise: EXERCISE 2.1 | Q: 1 | (Page 9)
For More Understanding

Explanation

The solution follows the method established in Example 1 and Example 2 of the provided text. First, the inverse function is converted to its trigonometric form. Second, the principal value branch range is identified from the text (Example 1 specifies the range for sin⁻¹ as [-π/2, π/2]). Finally, the negative angle is chosen because it falls within this specific range, consistent with the definition of principal value provided in Note 3.

Solution Steps

  1. Let sin⁡−1(−12)=y\sin^{-1}\left(-\frac{1}{2}\right) = y. Then, sin⁡y=−12\sin y = -\frac{1}{2}.

  2. We know that the range of the principal value branch of sin⁡−1\sin^{-1} is [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]. The value of an inverse trigonometric function which lies in the range of principal branch is called the principal value.

  3. We know that sin⁡(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. Therefore, using the property of negative angles, sin⁡(−π6)=−sin⁡(π6)=−12\sin\left(-\frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}.

  4. Since −π6-\frac{\pi}{6} lies within the range [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], the principal value of sin⁡−1(−12)\sin^{-1}\left(-\frac{1}{2}\right) is −π6-\frac{\pi}{6}.

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