SOLUTION: Find the exact values of sin 2𝜃, cos 2𝜃, and tan 2𝜃 for the given value of 𝜃. cot 𝜃 = 3/4;180° < 𝜃 < 270°

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Question 1201599: Find the exact values of sin 2𝜃, cos 2𝜃, and tan 2𝜃 for the given value of 𝜃.
cot 𝜃 = 3/4;180° < 𝜃 < 270°
Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
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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°.
~~~~~~~~~~~~~~~~~ 

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 =  =  ).


Hence, sin(a) = ;  cos(a) = .  The signs are  "-",  because we are in QIII.


Therefore

    sin(2a) = 2*sin(a)*cos(a) =  = .

    cos(2a) =  =  =  = .


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


Next,  tan(2a) =  =  = .

Solved.



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