Show that: cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000 = ½

By Ritesh|Updated : November 4th, 2022

Consider Cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000

To prove - cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000 = ½

Now consider:

LHS = cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000

= cos 240 + cos 550 + cos (1800 - 550) + cos (1800 + 240) + cos (2700 + 300)

As cos (1800 - x) = - cos x, cos (1800 + x) = - cos x

cos (2700 + x) = sin x

= cos 240 + cos 550 - cos 550 - cos 240 + sin 300

= sin 300

= ½

= RHS

The cosine (or cos) function of a triangle is the ratio of the adjacent side to the hypotenuse. The cosine function is one of the three main trigonometric functions and is itself the complement of the sine (cosine + sine).

Cosine Values

Cosine DegreesValues
cos 0°1
cos 60°

1/2

cos 30°

√3/2

cos 120°

-1/2

cos 45°

1/√2

cos 90°

0

cos 270°

0

cos 360°

1

cos 150°

-√3/2

cos 180°

-1

We can see in the above table that cos 120°, 150° and 180° are negative values where as cos 0°, 30°, etc. have positive values. The value will be positive in the first and the fourth quadrant for cos. Therefore, cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000 = ½

Summary:

Show that: cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000 = ½

It is proved that cos 240 + cos 550 + cos 1250 + cos 2040 + cos 3000 = ½. In a right triangle, cos is defined as the ratio of the length of the adjacent side to the length of the longest side, the hypotenuse.

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