Question

please solve it :-
$\cos(x ) \: = \: \cos(60) \: \cos(30) \: + \: \sin(60) \: \sin(30)$
​

Answer 1

$\large\underline{ \underline{ \sf \maltese{ \: Question:- }}}$

$\sf{ \cos(x ) = \bigg( \cos(60) \cos(30) \: + \: \sin(60) \sin(30) \bigg)}$

$\red{\boxed{ \sf \blue{ Find \: cos(x) \: ? }}}$

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$\large\underline{ \underline{ \sf \maltese{ \: Solution:- }}}$

$\qquad \quad {:} \longrightarrow \sf{\sf{ \cos(x ) \: = \: \cos(60) \: \cos(30) \: + \: \sin(60) \: \sin(30) }}$

$\quad {:} \longrightarrow \sf{\sf{ \sf{ \cos(x ) = \bigg( \frac{1}{2} \: \times \frac{ \sqrt{3} }{2} + \frac{ \sqrt{3} }{2} \: \times \frac{1}{2} \bigg)}}}$

$\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \frac{ \sqrt{3} }{4} \: + \: \frac{ \sqrt{3} }{4} }$

$\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \frac{ \sqrt{3} }{2} }$

$\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \cos(30) }$

$\qquad\quad {:} \longrightarrow \underline \red{\boxed{\sf{x \: = \: 30° }}}$

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$\begin{gathered}\begin{gathered}\qquad \therefore\: \sf{ \: \cos(x) = \underline {\underline{ \frac{ \sqrt{3} }{2} \: = \: 30° }}}\\\end{gathered}\end{gathered}$

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$\red{\boxed{ \sf \orange{ More \: To \: Learn\: :- }}}$

$\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}$

Answer 2

:⟶cos(x)=cos(60)cos(30)+sin(60)sin(30)

\quad {:} \longrightarrow \sf{\sf{ \sf{ \cos(x ) = \bigg( \frac{1}{2} \: \times \frac{ \sqrt{3} }{2} + \frac{ \sqrt{3} }{2} \: \times \frac{1}{2} \bigg)}}}:⟶cos(x)=(

2

1

×

2

3

+

2

3

×

2

1

)

\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \frac{ \sqrt{3} }{4} \: + \: \frac{ \sqrt{3} }{4} }:⟶cos(x)=

4

3

+

4

3

\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \frac{ \sqrt{3} }{2} }:⟶cos(x)=

2

3

\qquad \quad {:}\longrightarrow\sf{ \cos(x) \: = \: \cos(30) }:⟶cos(x)=cos(30)

\qquad\quad {:} \longrightarrow \underline \red{\boxed{\sf{x \: = \: 30° }}}:⟶

x=30°

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\begin{gathered}\begin{gathered}\begin{gathered}\qquad \therefore\: \sf{ \: \cos(x) = \underline {\underline{ \frac{ \sqrt{3} }{2} \: = \: 30° }}}\\\end{gathered}\end{gathered} \end{gathered}

∴cos(x)=

2

3

=30°

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