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Question: The principal value of f(x)=cos^(-1)⁡〖{cos⁡(-690°〗)} is...

The principal value of f(x)=cos^(-1)⁡〖{cos⁡(-690°〗)} is

A

π/3

B

2π/3

C

5π/3

D

π/6

Answer

π/6

Explanation

Solution

Here's how to find the principal value of cos^(-1){cos⁡(-690°)}:

  1. Use the even property of cosine:

    cos(-690°) = cos(690°)

  2. Reduce 690° modulo 360°:

    690° = 360° + 330° Therefore, cos(690°) = cos(330°)

  3. Recognize that:

    cos(330°) = cos(30°)

  4. The principal value of cos^(-1) is in the interval [0, π].

    Since 30° (or π/6) is in [0, π], we have: cos^(-1)(cos(-690°)) = π/6