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The area of right triangle ABC is 4, and the hypotenuse of AB is 12. Compute sin 2A

 Feb 15, 2022
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Let the sides of the triangle be  a  and  b  and the hypotenuse  c.

 

c = 12

 

Area  =  ½·a·b   --->   4  =  ½·a·b   --->    a·b   =  8   --->   b  =  8 / a

 

a2 + b2  =  c2   --->   a2 + (8/a)2  =  122   --->   a2 + 64/a2  =  144

                                                                --->   a4 + 64  =  144a2

                                                                --->   a4 - 14a2 + 64  =  0

 

Using the quadratic formula:  a2  =  [ 144 + sqrt(20480) ] / 2

                                   or         a2  =  [ 144 - sqrt(20480) ] / 2

 

Finding the square roots of the answers, either  a = 0.6677011      (approximately)

                                                                       or  a = 11.98140957  (approximately)

 

When a has either of these two values, b will have the other value.

 

From here, find the value of sinA =  0.6677011 / 12

                                       and sinB  =  11.98140957 / 12

(or vice versa, it makes no difference).

 

From here, use the formula for 2sin(x) = 2sin(x)cos(x)

to find your answer.

 Feb 15, 2022

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