Question

Find the value of

$\: \: \: \bull \: \sf \:sin(40 \degree \: + \theta \:)cos(10\degree \: + \theta \:) - cos(40\degree \: + \theta \:)sin(10\degree \: + \theta \:)$​

Answer 1

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Answer 2

Question:-

• Find the value of

$\odot \: \: \: \sf \:sin(40 \degree \: + \theta \:)cos(10\degree \: + \theta \:) - cos(40\degree \: + \theta \:)sin(10\degree \: + \theta \:)$

Given:-

• $\bf \:sin(40 \degree \: + \theta \:)cos(10\degree \: + \theta \:) - cos(40\degree \: + \theta \:)sin(10\degree \: + \theta \:)$

To find:-

• The value of given

Solution:-

$\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: \sin(40 \degree \: + \theta) \cos(10 \degree \: + \: \theta) - \cos(40 ^{ \sf \: o} + \theta) \sin( 40 \degree \: + \theta \: )$

$\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: : \implies \: \sin[ (40 \degree + \theta \: ) - (10 \degree + \theta)] \: \: \: \:$

We know that:-

$\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\: \sf \: \sin A \cos B - cosAsinB = sin(A - B)}$

$= \sf \: sin30 \degree \: (40^{o} + \theta - 10^{ \circ} - \theta)$

$= \sf \: sin30 \degree \:$

$\large{ \boxed{ = \frac{1}{2} }}$

$\large \dag \: \boxed{ \purple { \underline{ \purple{ \sf \: {Hence \: the \: required \: value \: is \: \frac{1}{2} }}}}}$

Step-by-step explanation:

Additional information:-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}$