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Question: A line makes angles of \(45 ^ { \circ }\)and \(60 ^ { \circ }\) with the positive axes of X and Y r...

A line makes angles of 45∘45 ^ { \circ }and 60∘60 ^ { \circ } with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is

A

30∘30 ^ { \circ } or 60∘60 ^ { \circ }

B

60∘60 ^ { \circ } or 90∘90 ^ { \circ }

C

90∘90 ^ { \circ } or 120∘120 ^ { \circ }

D

60∘60 ^ { \circ } or 120∘120 ^ { \circ }

Answer

60∘60 ^ { \circ } or 120∘120 ^ { \circ }

Explanation

Solution

Given

∵cos⁡2α+cos⁡2β+cos⁡2γ=1\because \cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1

∴cos⁡2γ=1−12−14=14⇒γ=60∘\therefore \cos ^ { 2 } \gamma = 1 - \frac { 1 } { 2 } - \frac { 1 } { 4 } = \frac { 1 } { 4 } \Rightarrow \gamma = 60 ^ { \circ } or 120∘120 ^ { \circ }.