The direction cosines of the line making angles π4, π3 and θ0<θ<π2 respectively with x, y and z axes, are

TS EAMCET · Maths · Three Dimensional Geometry

The direction cosines of the line making angles $\frac{π}{4}, \frac{π}{3}$ and $θ (0 < θ < \frac{π}{2})$ respectively with $x, y$ and $z$ axes, are

A $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$
B $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{\sqrt{3}}{2}$
C $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}}$
D $\frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$

Step-by-step Solution

Detailed explanation

Given angles π4 and π3, hence, direction cosines are cosπ4 and cosπ3 i.e. 12 and 12 ∴cos2α+cos2β+cos2γ=1 ⇒122 + 122 + cos2γ = 1…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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