The shooter in exercise 67 is assumed to be in the center of...

The shooter in exercise 67 is assumed to be in the center of the ice. Suppose that the line from the shooter to the center of the goal makes an angle of $\theta$ with center line. For the goalie to completely block the goal, he must stand $d$ feet away from the net where $d=D(1-w / 6 \cos \theta) .$ Show that for small angles, $d \approx D(1-w / 6)$

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Key Concepts

Small Angle Approximation

This concept involves approximating trigonometric functions for angles that are very small (measured in radians). When the angle is small, the cosine function can be approximated by 1, as higher order terms in its Taylor series are negligible. This idea is widely used in physics and engineering where small deviations allow the use of simpler linear models.

Taylor Series Expansion

The Taylor series expansion is a method used to represent functions as infinite sums of power series terms. For the cosine function, the expansion is cos(?) ? 1 - ?²/2 + ...; for small angles, the term ?²/2 and further terms are so small that cos(?) is effectively 1. This mathematical technique validates the small angle approximation and simplifies complex expressions.

Trigonometric Function Behavior

Understanding the behavior of trigonometric functions, particularly the cosine function, is crucial in problems involving angular deviations. Recognizing that cos(?) remains close to 1 for small angles allows for significant simplifications in equations and helps in approximating relationships in geometric configurations.

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