Question

Show that cos 24 + cos 55 + cos 125 + cos 204 + cos 300= 1/2

Answer 1

lhs =cos 24+cos 55+cos 125+cos 204+cos 300
= cos 24 +cos 55 +cos (180-55)+cos (180-24)+cos(270+30)
= cos 24 +cos 55-cos 55-cos24+sin30
=sin 30
=1/2
=rhs

Answer 2

Answer:

cos 24⁰ + cos 55⁰ + cos 125⁰ + cos 204⁰ + cos 300⁰ = 1/2 .

Step-by-step explanation:

• We know that :

$\cos( \alpha ) = \cos( \alpha ) \\ \cos(\pi - \alpha ) = - \cos( \alpha ) \\ \cos(\pi + \alpha ) = - \cos( \alpha ) \\ \cos(2\pi - \alpha ) = \cos( \alpha )$

• i.e. cosine of an angle is positive in 1st and 4th quadrant and negative in 2nd and 3rd quadrant.

Given that :

• L.H.S. = cos 24⁰ + cos 55⁰ + cos 125⁰ + cos 204⁰ + cos 300⁰

Solution:

• cos 125⁰ = cos ( 180⁰ - 55⁰ ) = - cos 55⁰
• cos 204⁰ = cos ( 180⁰ + 24⁰ ) = - cos 24⁰
• cos 300⁰ = cos ( 360⁰ - 300⁰ ) = cos 60⁰
• putting above values in L.H.S. , we get

$= \cos{24}^{0} + \cos {55}^{0} - \cos {55}^{0} - \cos {24}^{0} + \cos {60}^{0} \\ = \cos{60}^{0} \\ = \frac{1}{2}$

• L.H.S. = R.H.S.
• Hence, cos 24⁰ + cos 55⁰ + cos 125⁰ + cos 204⁰ + cos 300⁰ = 1/2 .