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Question

Using the measurements given in the following figure:
the value of $$\displaystyle \sin \Theta = \frac{m}{13}$$
value of m is 

186986_8047a88b72a4461aae959a2cf3ab4271.png


Solution

Draw a perpendicular from D on AB to meet at AB at E. Hence, DEBC is a rectangle.
Thus, $$DE = BC = 12$$
In $$\triangle DBC$$,
$$BD^2 = CD^2 + BC^2$$
$$13^2 = 12^2 + CD^2$$
$$CD^2 = 25$$
$$CD = 5$$
Now, $$BE = CD = 5$$
Thus, $$AE = AB - BE$$
$$AE = 14 - 5$$
$$AE = 9 $$
Now, In $$\triangle DBC$$
$$\sin \Phi = \frac{P}{H} = \frac{DC}{BD} = \frac{5}{13}$$
Now, In $$\triangle AED$$
$$\tan \Theta = \frac{DE}{AE} = \frac{9}{12} = \frac{3}{4}$$

Maths

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