If $8 tan A = 15$ , find $sin A - cos A .$

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Solution

Given,
$8 tan A = 15$

$tan A = \frac{15}{8}$

We know that,
$tan A = \frac{o p p o s i t e S i d e}{a d j a c e n t S i d e}$

From Pythagoras theorem,
$(H y p o t e n u s e)^{2} = (o p p o s i t e S i d e)^{2} + (a d j a c e n t S i d e)^{2}$
$(H y p o t e n u s e)^{2} = 15^{2} + 8^{2}$
$(H y p o t e n u s e)^{2} = 225 + 64 = 289$
$(H y p o t e n u s e) = 17$

$c o s A = \frac{A d j a c e n t S i d e}{H y p o t e n u s e} = \frac{8}{17}$
$s i n A = \frac{O p p o s i t e S i d e}{A d j a c e n t S i d e} = \frac{15}{17}$

Therefore,
$sin A - cos A$

$= \frac{15}{17} - \frac{8}{17}$

$= \frac{(15 - 8)}{17}$

$= \frac{7}{17}$

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