Evaluate the following determinant: \[\begin{vmatrix}\cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ\end{vmatrix}\]

\[∆ = \cos15^\circ\cos75^\circ - \sin15^\circ\sin75^\circ\] \[ = \cos15^\circ\cos75^\circ - \sin(90^\circ - 75^\circ)\sin(90^\circ - 15^\circ) \left[ \because \sin\left( 90^\circ - \theta \right) = \cos\theta \right]\] \[ = \cos15^\circ\cos75^\circ - \cos75^\circ\cos15^\circ\] \[ = \cos15^\circ\cos75^\circ - \cos15^\circ\cos75^\circ = 0\]

Evaluate the Following Determinant: ∣ ∣ ∣ Cos 15 ∘ Sin 15 ∘ Sin 75 ∘ Cos 75 ∘ ∣ ∣ ∣ - Mathematics

प्रश्न

Evaluate the following determinant:

$| \begin{matrix} \cos 15^{\circ} & \sin 15^{\circ} \\ \sin 75^{\circ} & \cos 75^{\circ} \end{matrix} |$

उत्तर

$∆ = \cos 15^{\circ} \cos 75^{\circ} - \sin 15^{\circ} \sin 75^{\circ}$
$= \cos 15^{\circ} \cos 75^{\circ} - \sin (90^{\circ} - 75^{\circ}) \sin (90^{\circ} - 15^{\circ}) [∵ \sin (90^{\circ} - θ) = \cos θ]$
$= \cos 15^{\circ} \cos 75^{\circ} - \cos 75^{\circ} \cos 15^{\circ}$
$= \cos 15^{\circ} \cos 75^{\circ} - \cos 15^{\circ} \cos 75^{\circ} = 0$

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Applications of Determinants and Matrices

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अध्याय 6: Determinants - Exercise 6.1 [पृष्ठ १०]

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अध्याय 6 Determinants
Exercise 6.1 | Q 2.3 | पृष्ठ १०

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Evaluate the Following Determinant: ∣ ∣ ∣ Cos 15 ∘ Sin 15 ∘ Sin 75 ∘ Cos 75 ∘ ∣ ∣ ∣ - Mathematics

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Evaluate the Following Determinant: ∣ ∣ ∣ Cos 15 ∘ Sin 15 ∘ Sin 75 ∘ Cos 75 ∘ ∣ ∣ ∣ - Mathematics

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