Counting Lines with Prescribed Angles to a Cube

★ 7/10trigonometry solid geometry Trigonometric Identities Casework

Problem

As shown in the figure, $P$ is a point outside the cube $ABCD$ - $A_1B_1C_1D_1$ . A line $l$ is drawn through $P$ ; let $\theta_1$ and $\theta_2$ denote the angles that $l$ makes with the lines $AC_1$ and $BC$ , respectively. Given that

\sin\left(\theta_1-50^\circ\right)=\cos\left(140^\circ-\theta_2\right)=\dfrac 12,

find the number of lines $l$ satisfying these conditions.

Answer

Solution

Difficulty	7/10
Topics	trigonometry, solid geometry, Trigonometric Identities, Casework

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