Question 1201599
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Find the exact values of sin 2a, cos 2a, and tan 2a for the given value of a.
cot(a) = 3/4;180° < a < 270°.
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<pre>
Since  180° < a < 270°,  angle  "a"  is in the 3rd quadrant, QIII.

 
From the definition of the cot-function, it is the ratio of the attached leg to the opposite leg.

    So, the attached leg to angle "a" of the right angled triangle is 3 units long horizontally, opposite to x-axis;
        the opposite leg to angle "a" of the right angled triangle is 4 units long vertically,   opposite to y-axis.


The hypotenuse is 5 units long  ( 5 = {{{sqrt(3^2+4^2)}}} = {{{sqrt(25)}}} ).


Hence, sin(a) = {{{-4/5}}};  cos(a) = {{{-3/5)}}}.  The signs are  "-",  because we are in QIII.


Therefore

    sin(2a) = 2*sin(a)*cos(a) = {{{2*(-4/5)*(-3/5)}}} = {{{24/25}}}.

    cos(2a) = {{{cos^2(a)-sin^2(a)}}} = {{{(-3/5)^2 - (-4/5)^2}}} = {{{9/25 - 16/25}}} = {{{-7/25}}}.


By the way, it means that angle "2a" is in QII.


Next,  tan(2a) = {{{sin(2a)/cos(2a)}}} = {{{((24/25))/(-7/25))}}} = {{{-24/7}}}.
</pre>

Solved.