Question

Now, find the angle θ4 shown in the figure (Figure 3) in terms of θ1 and α.

Express your answer in degrees in terms of θ1 and α.

Answer 1

Concepts and reason

The concepts related to this problem are the law of reflection and the triangle rule that related to the sum of the angles of a triangle.

Initially, find the relation between the anglesand using the condition that the total angle made by the normal with the surface is equal to. Later, use the law of reflection and equate the anglesand. Then, we get the relation between the anglesand. Later, use the condition that the sum of three angles of a triangle is equal to and find a relation between,, and . Later, eliminate the termfrom this relation by using the relation betweenand. Then, the final equation contains the terms,, and. Finally, rearrange the equation for. This equation represents the required result.

Fundamentals

The imaginary line which is normal to the common surface of two transparent media is called normal. The angle between the incidence ray and the normal is called angle of incidence and the angle between the reflected ray and the normal is called angle of reflection.

From the law of reflection, the angle of incidence is equal to the angle of reflection. In the figure, represents the angle of incidence at first incidence and represents the corresponding angle of reflection. The terms,, andrepresents the three angles of the triangle which is formed by one ray and two surfaces.

Use the law of reflection and find the relation betweenand.

From the figure, the total angle made by the normal to the horizontal surface is equal to.

Substitutefor.

From the rules of triangles, the sum of three angles of a triangle is equal to.

Use the equationin this equation, and rearrange the equation for.

Ans:

The expression for anglein terms of the angleand the angleis.