Question 3 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: cosec⁻¹(2)

Correct Answer

Let cosec⁡−1(2)=y\cosec^{-1}(2) = y. Then, cosec⁡y=2\cosec y = 2.

We know that cosec⁡(π6)=2\cosec\left(\frac{\pi}{6}\right) = 2. According to the context, the principal value branch of cosec⁡−1\cosec^{-1} is the interval [−π2,π2]−{0}\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}.

Since π6\frac{\pi}{6} lies within the range [−π2,π2]\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] and is not zero, it satisfies the condition for the principal value.

Hence, the principal value of cosec⁡−1(2)\cosec^{-1}(2) is π6\frac{\pi}{6}.

Exercise: EXERCISE 2.1 | Q: 3 | (Page 9)
For More Understanding

Explanation

The solution follows the methodology established in Example 2 of the provided text. We define the inverse function in terms of a variable y, convert it to its direct trigonometric form, and identify the angle. The context explicitly provides the range for the principal value branch of cosec⁻¹ as [−π/2, π/2] − {0}. Since π/6 falls within this defined range and satisfies the equation cosec(π/6) = 2, it is the correct principal value.

Solution Steps

  1. Step 1: Let cosec⁡−1(2)=y\cosec^{-1}(2) = y. This implies cosec⁡y=2\cosec y = 2.

  2. Step 2: Identify the angle where cosec⁡y=2\cosec y = 2. We know cosec⁡(π6)=2\cosec\left(\frac{\pi}{6}\right) = 2.

  3. Step 3: Verify the range. The context states the principal range for cosec⁡−1\cosec^{-1} is [−π2,π2]−{0}\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}.

  4. Step 4: Since π6\frac{\pi}{6} lies in the range [−π2,π2]−{0}\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] - \{0\}, the principal value is π6\frac{\pi}{6}.

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