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Verizon [17]
2 years ago
10

The line makes angles α, β and γ with x-axia and z-axis respectively then cos 2α + cos 2β + cos 2γ is equal to

Mathematics
1 answer:
Elis [28]2 years ago
4 0

Step-by-step explanation:

\large\underline{\sf{Solution-}}

Given that lines makes an angle α, β, γ with x - axis, y - axis and z - axis respectively.

So, By definition of direction cosines,

\rm :\longmapsto\:l = cos \alpha

\rm :\longmapsto\:m = cos \beta

\rm :\longmapsto\:n = cos \gamma

So,

\rm :\longmapsto\: {l}^{2} +  {m}^{2}  +  {n}^{2}  = 1

\rm :\longmapsto\: {cos}^{2} \alpha  +  {cos}^{2} \beta  +  {cos}^{2} \gamma  = 1

On multiply by 2 on both sides we get

\rm :\longmapsto\: 2{cos}^{2} \alpha  +  2{cos}^{2} \beta  + 2 {cos}^{2} \gamma  = 2

can be further rewritten as

\rm :\longmapsto\: 2{cos}^{2} \alpha  - 1 + 1 +  2{cos}^{2} \beta  - 1 + 1 + 2 {cos}^{2} \gamma  - 1 + 1 = 2

\rm :\longmapsto\: (2{cos}^{2} \alpha  - 1)+ (2{cos}^{2} \beta  - 1)+ (2 {cos}^{2} \gamma  - 1) + 3= 2

\rm :\longmapsto\:cos2 \alpha  + cos2 \beta  + cos2 \gamma   + 3= 2

\red{ \bigg\{  \sf \: \because \: cos2x =  {2cos}^{2}x - 1  \bigg\}}

\rm :\longmapsto\:cos2 \alpha  + cos2 \beta  + cos2 \gamma= 2 - 3

\rm :\longmapsto\:cos2 \alpha  + cos2 \beta  + cos2 \gamma=  - 1

Hence,

\bf\implies \:\boxed{\tt{  \: cos2 \alpha  + cos2 \beta  + cos2 \gamma  =  - 1 \: }}

So, option (d) is correct.

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

<h3>MORE TO KNOW </h3>

Direction cosines of a line segment is defined as the cosines of the angle which a line makes with the positive direction of respective axis.

The scalar components of unit vector always give direction cosines.

The scalar components of a vector gives direction ratios.

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Now consider the remaining 26 odd numbers:

{1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99}

which can be written in terms of their prime factors as:

{1, 3*3, 3*5, 3*7, 5*5,3*3*3, 3*11,5*7, 3*13, 2*2*3*3, 7*7, 3*17, 5*11 , 3*19,3*21, 5*13, 3*23,3*5*5, 7*11, 3*3*3*3, 5*17, 3*29, 7*13, 3*31, 5*19, 3*3*11}

 

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