Question 10 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: cosec⁻¹(-2\sqrt{2})

Correct Answer

Let cosec⁻¹(-2\sqrt{2}) = y. Then, cosec y = -2\sqrt{2} = -cosec(π/4) = cosec(-π/4). Since the principal value branch of cosec⁻¹ is [-π/2, π/2] - {0}, and -π/4 lies in this interval, the principal value of cosec⁻¹(-2\sqrt{2}) is -π/4.

Exercise: EXERCISE 2.1 | Q: 10 | (Page 10)
For More Understanding

Explanation

This question tests understanding of the principal value branch of inverse cosecant function. From the textbook context, the principal branch of cosec⁻¹ has range [-π/2, π/2] - {0}. The student needs to express -2\sqrt{2} as cosec of an angle within this range. Since cosec(π/4) = 2\sqrt{2}, we get -2\sqrt{2} = -cosec(π/4) = cosec(-π/4). The angle -π/4 falls within the principal range, making it the valid answer.

Solution Steps

  1. Step 1: Let cosec⁻¹(-2\sqrt{2}) = y, then cosec y = -2\sqrt{2}

  2. Step 2: Express -2\sqrt{2} as cosec of standard angle: cosec(π/4) = 2\sqrt{2}, so -2\sqrt{2} = -cosec(π/4) = cosec(-π/4)

  3. Step 3: Verify -π/4 lies in principal range [-π/2, π/2] - {0}

  4. Step 4: Conclude principal value is -π/4

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