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A line makes α/2, β/2, γ/2 angles with positive direction of coordinate axes then cosα + cosβ + cosγ equals-
Correct answer is '1'. Can you explain this answer?
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A line makes α/2, β/2, γ/2 angles with positive direc...
Given Information:
A line makes α/2, β/2, γ/2 angles with the positive direction of the coordinate axes.
Explanation:
When a line makes angles of α/2, β/2, γ/2 with the positive direction of the coordinate axes, the direction cosines of the line are given by cos(α/2), cos(β/2), and cos(γ/2).
Relationship between Direction Cosines and Cosines of Angles:
The relationship between the direction cosines (l, m, n) and the cosines of angles (α, β, γ) is given by:
l = cos(α)
m = cos(β)
n = cos(γ)
Expression to Evaluate:
We need to find the value of cosα + cosβ + cosγ.
Substitute l = cos(α/2), m = cos(β/2), and n = cos(γ/2) into the expression:
cosα + cosβ + cosγ = 2cos(α/2)cos(β/2)cos(γ/2)
Using Trigonometric Identity:
cos(A) + cos(B) + cos(C) = 2cos((A+B)/2)cos((B+C)/2)cos((C+A)/2)
Substitute A = α, B = β, C = γ into the identity:
cosα + cosβ + cosγ = 2cos((α + β)/2)cos((β + γ)/2)cos((γ + α)/2)
cosα + cosβ + cosγ = 2cos(180/2)cos(180/2)cos(180/2)
cosα + cosβ + cosγ = 2cos(90)cos(90)cos(90)
cosα + cosβ + cosγ = 2(0)(0)(0) = 0
Therefore, cosα + cosβ + cosγ equals 0.
A line makes α/2, β/2, γ/2 angles with the positive direction of the coordinate axes.
Explanation:
When a line makes angles of α/2, β/2, γ/2 with the positive direction of the coordinate axes, the direction cosines of the line are given by cos(α/2), cos(β/2), and cos(γ/2).
Relationship between Direction Cosines and Cosines of Angles:
The relationship between the direction cosines (l, m, n) and the cosines of angles (α, β, γ) is given by:
l = cos(α)
m = cos(β)
n = cos(γ)
Expression to Evaluate:
We need to find the value of cosα + cosβ + cosγ.
Substitute l = cos(α/2), m = cos(β/2), and n = cos(γ/2) into the expression:
cosα + cosβ + cosγ = 2cos(α/2)cos(β/2)cos(γ/2)
Using Trigonometric Identity:
cos(A) + cos(B) + cos(C) = 2cos((A+B)/2)cos((B+C)/2)cos((C+A)/2)
Substitute A = α, B = β, C = γ into the identity:
cosα + cosβ + cosγ = 2cos((α + β)/2)cos((β + γ)/2)cos((γ + α)/2)
cosα + cosβ + cosγ = 2cos(180/2)cos(180/2)cos(180/2)
cosα + cosβ + cosγ = 2cos(90)cos(90)cos(90)
cosα + cosβ + cosγ = 2(0)(0)(0) = 0
Therefore, cosα + cosβ + cosγ equals 0.
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A line makes α/2, β/2, γ/2 angles with positive direction of coordinate axes then cosα + cosβ + cosγ equals-Correct answer is '1'. Can you explain this answer?
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A line makes α/2, β/2, γ/2 angles with positive direction of coordinate axes then cosα + cosβ + cosγ equals-Correct answer is '1'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A line makes α/2, β/2, γ/2 angles with positive direction of coordinate axes then cosα + cosβ + cosγ equals-Correct answer is '1'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A line makes α/2, β/2, γ/2 angles with positive direction of coordinate axes then cosα + cosβ + cosγ equals-Correct answer is '1'. Can you explain this answer?.
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