If 2 π ≤ x ≤ 4 3 π , then cos − 1 ( 13 12 cos x + 13 5 sin x ) is equal to

13 12 cos x + 13 5 sin x Let tan α = 12 5 , α ∈ ( 0 , 2 π ) ⇒ sin α = 13 5 , cos α = 13 12 ⇒ 13 12 cos x + 13 5 sin x = cos α cos x + sin α sin x = cos ( x − α ) ⇒ cos − 1 [ cos ( x − α )] = x − α = x − tan − 1 ( 12 5 )

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Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

MathematicsTrigonometric Ratios & IdentitiesAddition, Multiplication, Subtraction formula based,Multiple/Submultiple anglesMedium2 minPYQ_2025

MathematicsMediumsingle choice

If $\frac{π}{2} \leq x \leq \frac{3 π}{4}$ , then $cos^{- 1} (\frac{12}{13} cos x + \frac{5}{13} sin x)$ is equal to

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Solution:

$\frac{12}{13} cos x + \frac{5}{13} sin x Let tan α = \frac{5}{12}, α \in (0, \frac{π}{2}) \Rightarrow sin α = \frac{5}{13}, cos α = \frac{12}{13}$
$\Rightarrow \frac{12}{13} cos x + \frac{5}{13} sin x = cos α cos x + sin α sin x = cos (x - α) \Rightarrow cos^{- 1} [cos (x - α)] = x - α = x - tan^{- 1} (\frac{5}{12})$

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Addition, Multiplication, Subtraction formula based,Multiple/Submultiple angles

⏱ 2mℹ️ Source: PYQ_2025

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