Question

in triangle ABC right angled at B if tan a = root3 find value of i) Sin A cos C + cos A sin C ii) cos A cos C - sin A sin C

Answer 1

Answer:

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Answer 2

Given,

Triangle ABC is right angle triangle.

$\[\angle B = 90^\circ \]$

$\[\tan A = \sqrt 3 \]$

Solution,

Calculate the value of angles,

$\[\begin{array}{l}\tan A = \sqrt 3 \\ \Rightarrow \tan A = \tan 60^\circ \\ \Rightarrow \angle A = 60^\circ \end{array}\]$

Know that for a triangle,

$\[\begin{array}{l}\angle A + \angle B + \angle C = 180^\circ \\ \Rightarrow 60^\circ + 90^\circ + \angle C = 180^\circ \\ \Rightarrow \angle C = 30^\circ \end{array}\]$

$\[\begin{array}{l}\left( i \right)\sin A\cos C + \cos A\sin C\\ = \sin \left( {A + C} \right)\\ = \sin \left( {60^\circ + 30^\circ } \right)\\ = \sin 90^\circ \\ = 1\end{array}\]$

Hence the value is 1.

$\[\begin{array}{l}\left( {ii} \right)\cos A\cos C - \sin A\sin C\\ = \cos \left( {A + C} \right)\\ = \cos \left( {60^\circ + 30^\circ } \right)\\ = \cos 90^\circ \\ = 0\end{array}\]$

Hence the value is 0.