Question 1204118
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If tan(x) = −40/9 and x is in quadrant IV, with 0° ≤ x < 360°, find the exact values of the expressions without solving for x.

(a) sin(x/2)

(b) cos(x/2)

(c) tan(x/2)
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<pre>
If tan(x) = −40/9 and x is in quadrant IV, then it means that 

    the opposite leg is 40 units, the adjacent leg is 9 units and the hypotenuse is  {{{sqrt(40^2 + 9^2)}}} = {{{sqrt(1681)}}} = 41 unit.

It implies that  cos(x) = {{{9/41}}}.


Next, use the formulas for half argument


    {{{sin(x/2)}}} = {{{-sqrt((1 - cos(x))/2)}}} = {{{-sqrt((1-9/41)/2)}}} = {{{-sqrt(16/41)}}} = {{{-4/sqrt(41)}}};


    {{{cos(x/2)}}} = {{{sqrt((1 + cos(x))/2)}}} = {{{sqrt((1+9/41)/2)}}} = {{{sqrt(25/41)}}} = {{{5/sqrt(41)}}};


    {{{tan(x/2)}}} = {{{-4/5}}}.
</pre>

Solved.