Let α, β be such that π<α-β<3π. If α+β=-21/65 and α+β=-27/65, then the value of α-β/2 is - Mathematics MCQ AIEEE 2004 2004 | ExamTest.live

hardMCQAIEEE 20042004Mathematics

1 attempts0% success rate3 marks (−1)

Let α, β be such that π < α - β < 3 π . If sin α + sin β = - \frac{21}{65} and cos α + cos β = - \frac{27}{65}, then the value of cos \frac{α - β}{2} is

Solution & Step-by-step Explanation

Given: sin α + sin β = - \frac{21}{65} cos α + cos β = - \frac{27}{65} . Squaring and adding both equations: (sin^{2} α + cos^{2} α) + (sin^{2} β + cos^{2} β) + 2 (sin α sin β + cos α cos β) = (\frac{21}{65})^{2} + (\frac{27}{65})^{2} 1 + 1 + 2 cos (α - β) = \frac{441 + 729}{6 5 ^{2}} = \frac{1170}{4225} . 2 [1 + cos (α - β)] = \frac{1170}{4225} Using 1 + cos θ = 2 cos^{2} \frac{θ}{2} : 4 cos^{2} \frac{α - β}{2} = \frac{1170}{4225} = \frac{18}{65} . cos^{2} \frac{α - β}{2} = \frac{18}{4 \times 65} = \frac{9}{130} . cos \frac{α - β}{2} = \pm \frac{3}{130} . Since π < α - β < 3 π, then \frac{π}{2} < \frac{α - β}{2} < \frac{3 π}{2} . In this interval, the cosine function is negative. cos \frac{α - β}{2} = - \frac{3}{130} .

Try it yourself before checking the explanation above.

Let α, β be such that π < α - β < 3 π . If sin α + sin β = - \frac{21}{65} and cos α + cos β = - \frac{27}{65}, then the value of cos \frac{α - β}{2} is

- \frac{3}{130}

\frac{3}{130}

\frac{6}{65}

- \frac{6}{65}

Practice with timed mock tests and track your performance across Mathematics.