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Maths-

General

Easy

Question

$5 cos space theta plus 3 cos space open parentheses theta minus pi over 3 close parentheses plus 8 element of$

$[- 7, 1]$
$[1, 8]$
$[1, 15]$
$[2, 15]$

hint Hint:

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

The correct answer is: $left square bracket 1 comma 15 right square bracket$

$5 cos space theta plus 3 cos space open parentheses theta minus pi over 3 close parentheses plus 8 equals 5 space cos theta plus 3 open parentheses cos theta. cos straight pi over 3 plus sin theta. sin straight pi over 3 close parentheses plus 8 equals 5 cos theta plus 3 open parentheses fraction numerator cos theta over denominator 2 end fraction plus fraction numerator square root of 3 sin theta over denominator 2 end fraction close parentheses plus 8 equals 5 cos theta plus 3 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 equals 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 minus square root of a squared plus b squared end root less or equal than a space 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 fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses squared plus open parentheses 13 over 2 close parentheses squared end root less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than square root of open parentheses fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses squared plus open parentheses 13 over 2 close parentheses squared end root rightwards double arrow negative 7 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than 7 rightwards double arrow negative 7 plus 8 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta plus 8 less or equal than 7 plus 8 rightwards double arrow 1 less or equal than 13 over 2 cos theta plus fraction numerator 3 square root of 3 over denominator 2 end fraction sin theta less or equal than 15 f left parenthesis x right parenthesis element of open square brackets 1 comma 15 close square brackets$

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