Question 5 of 42beginner🔧 ApplyShort Answer2 marks

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

Correct Answer

2π/3

Exercise: EXERCISE 2.1 | Q: 5 | (Page 9)
For More Understanding

Explanation

The solution follows the standard method for finding principal values of inverse trigonometric functions. It identifies the principal value branch for cos⁻¹ as [0, π] from the provided text. It then uses the trigonometric property that cos(π - θ) = -cos(θ) to find the specific angle within that range.

Solution Steps

  1. Step 1: Let y=cos⁡−1(−1/2)y = \cos^{-1}(-1/2). Then, cos⁡y=−1/2\cos y = -1/2.

  2. Step 2: We know that cos⁡(π/3)=1/2\cos(\pi/3) = 1/2. Using the identity cos⁡(π−θ)=−cos⁡(θ)\cos(\pi - \theta) = -\cos(\theta), we can write −1/2-1/2 as −cos⁡(π/3)=cos⁡(π−π/3)=cos⁡(2π/3)-\cos(\pi/3) = \cos(\pi - \pi/3) = \cos(2\pi/3).

  3. Step 3: From the context, the range of the principal value branch of cos⁡−1\cos^{-1} is [0,π][0, \pi].

  4. Step 4: Since 2π/32\pi/3 lies in the interval [0,π][0, \pi], the principal value of cos⁡−1(−1/2)\cos^{-1}(-1/2) is 2π/32\pi/3.

PreviousNext