Question

Prove that:
cos20 + cos100 + cos140=0

Class 11

Trigonometry

Thank you ​

Answer 1

Answer:

Explanation:

We use the formula cos(A) + cos(120-A) + cos(120+A) = 0

Proof=

L.H.S= cosA + cos(120-A) + cos(120+A)

we know that formula

( cos C+ cosD = 2cos (C+D)/2.cos (C-D) /2)

=> cosA+2cos120 cos-A

=> cosA+2cos (180-60) cosA

=> cosA+2(-cos60) cosA

cosa - 2 \times \frac{1}{2}cosa

=> cosA-cosA

=> 0

In this question, cos20 + cos100 + cos140

=>cos20 + cos(120-20) + cos(120+20)

=>0

Answer 2

Answer:-

Explanation:-

Take LHS,

(Cos20 + Cos100) + Cos140 = 0

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using identity:-

Cos(x+ y) = Cos(x)Cos(y) - Sin(x)Sin(y)

Cos(x - y) = Cos(x)Cos(y) + Sin(x)Sin(y)

________________________

We know that :-

140 = 120 + 20

and

100 = 120 - 20

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=> Cos140 + Cos 100 + Cos20

=> Cos(120) Cos(20) - Sin(120) Sin(20) + Cos(120) Cos(20) = Sin(120) Sin (20) + Cos(20)

=> Cos(20) [2Cos(120) + 1]

= Cos(20) [(2) (-1/2) +1]

=> Cos(20) [0]

=> 0

HENCE PROVED

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