NTA Abhyas 2020
Exam NTA Abhyas
Year 2020
Subject Mathematics
Chapter Trigonometric Equations NTA Abhyas 2020 · Mathematics · Trigonometric Equations
If 0 < α, β < π and cos α + cos β - (α + β) = (3)/(2) , then the...
If
0 < α , β < π 0 < \alpha, \beta < \pi 0 < α , β < π and
c o s α + c o s β − cos ( α + β ) = 3 2 cos \alpha + cos \beta - \cos \left(\alpha + \beta\right) = \dfrac{3}{2} cos α + cos β − cos ( α + β ) = 2 3 , then the value of
3 s i n α + c o s α \sqrt{3}sin \alpha + cos \alpha 3 s in α + cos α is equal to
Step-by-step solution
2 cos ( α + β 2 ) cos ( α − β 2 ) − ( 2 c o s 2 ( α + β 2 ) − 1 ) = 3 2 2\cos \left(\dfrac{\alpha + \beta}{2}\right)\cos \left(\dfrac{\alpha - \beta}{2}\right) - \left(2co{s}^{2}\left(\dfrac{\alpha + \beta}{2}\right) - 1\right) = \dfrac{3}{2} 2 cos ( 2 α + β ) cos ( 2 α − β ) − ( 2 co s 2 ( 2 α + β ) − 1 ) = 2 3 ⇒ 4 c o s 2 ( α + β 2 ) − 4 cos ( α + β 2 ) cos ( α − β 2 ) + 1 = 0 ⇒ 4co{s}^{2}\left(\dfrac{\alpha + \beta}{2}\right) - 4\cos \left(\dfrac{\alpha + \beta}{2}\right)\cos \left(\dfrac{\alpha - \beta}{2}\right) + 1 = 0 ⇒ 4 co s 2 ( 2 α + β ) − 4 cos ( 2 α + β ) cos ( 2 α − β ) + 1 = 0 ⇒ ( 2 cos ( α + β 2 ) − cos ( α − β 2 ) ) 2 + s i n 2 ( α − β 2 ) = 0 ⇒ {\left(2\cos \left(\dfrac{\alpha + \beta}{2}\right) - \cos \left(\dfrac{\alpha - \beta}{2}\right)\right)}^{2} + si{n}^{2}\left(\dfrac{\alpha - \beta}{2}\right) = 0 ⇒ ( 2 cos ( 2 α + β ) − cos ( 2 α − β ) ) 2 + s i n 2 ( 2 α − β ) = 0 ⇒ 2 c o s α + β 2 = c o s α − β 2 ⇒ 2cos\dfrac{\alpha + \beta}{2} = cos\dfrac{\alpha - \beta}{2} ⇒ 2 cos 2 α + β = cos 2 α − β and
sin ( α − β 2 ) = 0 \sin \left(\dfrac{\alpha - \beta}{2}\right) = 0 sin ( 2 α − β ) = 0 ⇒ α = β ⇒ \alpha = \beta ⇒ α = β and
2 c o s α = 1 ⇒ α = π 3 ( 0 < α , β < π ) 2cos \alpha = 1 ⇒ \alpha = \dfrac{\pi}{3}\left(0 < \alpha, \beta < \pi\right) 2 cos α = 1 ⇒ α = 3 π ( 0 < α , β < π ) So,
s i n α = 3 2 sin \alpha = \dfrac{\sqrt{3}}{2} s in α = 2 3 and
c o s α = 1 2 cos \alpha = \dfrac{1}{2} cos α = 2 1 ⇒ 3 s i n α + c o s α = 2 ⇒ \sqrt{3}sin \alpha + cos \alpha = 2 ⇒ 3 s in α + cos α = 2
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