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Maths-

General

Easy

Question

For all values of $theta$ , the values of $3 minus cos space theta plus cos space open parentheses theta plus pi over 3 close parentheses$ lie in the interval

$[- 2, 3]$
$[- 2, 1]$
$[2, 4]$
$[1, 5]$

hint Hint:

In this question we have to find the range of the given equation. For this we will first simplify the equation using some trigonometric identities then using $negative square root of a squared plus b squared end root less or equal than a sin x plus b cos x less or equal than square root of a squared plus b squared end root$ we will find the range.

The correct answer is: $left square bracket 2 comma 4 right square bracket$

$3 minus cos space theta plus cos space open parentheses theta plus pi over 3 close parentheses equals 3 minus cos theta plus open parentheses cos theta. cos straight pi over 3 minus sin theta. sin straight pi over 3 close parentheses equals 3 minus cos theta plus 1 half cos theta minus fraction numerator square root of 3 over denominator 2 end fraction sin theta equals 3 minus 1 half cos theta minus fraction numerator square root of 3 over denominator 2 end fraction sin theta equals 3 minus 1 half open parentheses cos theta plus square root of 3 sin theta close parentheses N o w comma space minus square root of a squared plus b squared end root less or equal than a sin x plus b cos x less or equal than square root of a squared plus b squared end root rightwards double arrow negative square root of open parentheses negative square root of 3 close parentheses squared plus 1 squared end root less or equal than square root of 3 sin theta plus cos theta less or equal than square root of open parentheses negative square root of 3 close parentheses squared plus 1 squared end root rightwards double arrow negative 2 less or equal than square root of 3 sin theta plus cos theta less or equal than 2 rightwards double arrow 1 greater or equal than fraction numerator negative 1 over denominator 2 end fraction open parentheses square root of 3 sin theta plus cos theta close parentheses greater or equal than negative 1 rightwards double arrow 1 plus 3 greater or equal than 3 minus 1 half open parentheses square root of 3 sin theta plus cos theta close parentheses greater or equal than negative 1 plus 3 rightwards double arrow 4 greater or equal than 3 minus 1 half open parentheses square root of 3 sin theta plus cos theta close parentheses greater or equal than 2 S o comma space 3 minus cos space theta plus cos space open parentheses theta plus pi over 3 close parentheses element of open square brackets 2 comma 4 close square brackets$

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