Chapter 7.2, Problem 12CFU

To determine

To verify: that the given formulas of illuminance $\left(E\right)$ are equivalent.

Answer to Problem 12CFU

The right side of equation is transformed into left hand side.

Explanation of Solution

Given information:

The formulas of illuminance $\left(E\right)$ are shown below.

  $\begin{array}{c}E=\frac{I\cos \theta }{R^2}\\ E=\frac{I\cot \theta }{R^2\csc \theta }\end{array}$

Calculation:

The verification of the equivalence is shown below.

  $\begin{array}{c}\frac{I\cos \theta }{R^2}=\frac{I\cot \theta }{R^2\csc \theta }\\ =\frac{I\frac{\cos \theta }{\sin \theta }}{R^2\frac{1}{\sin \theta }}\left[\begin{array}{l}\text{Quotient identity of cotangent}\\ \text{Reciprocal identity of cosecant}\end{array}\right]\\ =\frac{\frac{I\cos \theta }{\sin \theta }}{\frac{R^2}{\sin \theta }}\\ =\frac{I\cos \theta }{R^2}\\ LHS=RHS\end{array}$

Since right side of equation is transformed into left hand side. So these formulas of $E$ are equivalent.