The given expression is5 c o s θ + 3 ( c o s θ c o s 60 o − s i n θ s i n 60 o ) + 3 5cos \theta + 3\left(cos \theta cos {60}^{o} - sin \theta sin {60}^{o}\right) + 3 5 cos θ + 3 ( cos θ cos 60 o − s in θ s in 60 o ) + 3 = 5 c o s θ + 3 c o s θ 1 2 − 3 s i n θ 3 2 + 3 = 5cos \theta + 3cos \theta\dfrac{1}{2} - 3sin \theta\dfrac{\sqrt{3}}{2} + 3 = 5 cos θ + 3 cos θ 2 1 − 3 s in θ 2 3 + 3 = c o s θ ( 5 + 3 2 ) + s i n θ ( − 3 3 2 ) + 3 = cos \theta\left(5 + \dfrac{3}{2}\right) + sin \theta\left(- \dfrac{3\sqrt{3}}{2}\right) + 3 = cos θ ( 5 + 2 3 ) + s in θ ( − 2 3 3 ) + 3 = 13 2 c o s θ − 3 3 2 s i n θ + 3 = \dfrac{13}{2}cos \theta - \dfrac{3\sqrt{3}}{2}sin \theta + 3 = 2 13 cos θ − 2 3 3 s in θ + 3 13 2 c o s θ − 3 3 2 s i n θ ∈ [ − k , k ] \dfrac{13}{2}cos \theta - \dfrac{3\sqrt{3}}{2}sin \theta \in \left[- k, k\right] 2 13 cos θ − 2 3 3 s in θ ∈ [ − k , k ] k = ( 13 2 ) 2 + ( − 3 3 2 ) 2 = 169 4 + 27 4 = 14 2 = 7 k = \sqrt{{\left(\dfrac{13}{2}\right)}^{2} + {\left(- \dfrac{3\sqrt{3}}{2}\right)}^{2}} = \sqrt{\dfrac{169}{4} + \dfrac{27}{4}} = \dfrac{14}{2} = 7 k = ( 2 13 ) 2 + ( − 2 3 3 ) 2 = 4 169 + 4 27 = 2 14 = 7 ⇒ 13 2 c o s θ − 3 3 2 s i n θ ∈ [ − 7 , 7 ] ⇒ \dfrac{13}{2}cos \theta - \dfrac{3\sqrt{3}}{2}sin \theta \in \left[- 7,7\right] ⇒ 2 13 cos θ − 2 3 3 s in θ ∈ [ − 7 , 7 ] ⇒ 13 2 c o s θ − 3 3 2 s i n θ + 3 ∈ [ − 4 , 10 ] ⇒ \dfrac{13}{2}cos \theta - \dfrac{3\sqrt{3}}{2}sin \theta + 3 \in \left[- 4,10\right] ⇒ 2 13 cos θ − 2 3 3 s in θ + 3 ∈ [ − 4 , 10 ] ⇒ a = − 4 ⇒ a = - 4 ⇒ a = − 4 b = 10 b = 10 b = 10
⇒ a + b = 6 ⇒ a + b = 6 ⇒ a + b = 6
