Question #245a7

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Answer 1 · 2017-06-24T09:59:15

#theta approx pi n - 81.9^(@), pi n + 71.6^(@)#; #n in NN#

We have: #frac(4 tan(theta))(21 - tan^(2)(theta)) = 1#

#Rightarrow 4 tan(theta) = 21 - tan^(2)(theta)#

#Rightarrow tan^(2)(theta) + 4 tan(theta) - 21 = 0#

Let's apply the quadratic formula:

#Rightarrow tan(theta) = frac(- 4 pm sqrt(4^(2) - 4(1)(- 21)))(2(1))#

#Rightarrow tan(theta) = frac(- 4 pm sqrt(16 + 84))(2)#

#Rightarrow tan(theta) = frac(- 4 pm sqrt(100))(2)#

#Rightarrow tan(theta) = frac(- 4 pm 10)(2)#

#Rightarrow tan(theta) = - 2 pm 5#

#Rightarrow tan(theta) = - 7, 3#

#therefore theta = pi n - arctan(7), pi n + arctan(3)#

#Rightarrow theta approx pi n - 81.9^(@), pi n + 71.6^(@)#; #n in NN#