Piston Engines See the figure. In a certain piston engine,...

Piston Engines See the figure. In a certain piston engine, the distance $x$ (in inches) from the center of the drive shaft to the head of the piston is modeled by the function $$ x(\theta)=\cos \theta+\sqrt{16+0.5\left(2 \cos ^{2} \theta-1\right)} $$ where $\theta$ is the angle between the crank and the path of the piston head. Find $x$ when $\theta=30^{\circ}$ and when $\theta=45^{\circ}$.

Piston Engines See the figure. In a certain piston engine, the distance $x$ (in inches) from the center of the drive shaft to the head of the piston is modeled by the function
$$
x(\theta)=\cos \theta+\sqrt{16+0.5\left(2 \cos ^{2} \theta-1\right)}
$$
where $\theta$ is the angle between the crank and the path of the piston head. Find $x$ when $\theta=30^{\circ}$ and when $\theta=45^{\circ}$.

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Key Concepts

Trigonometric Functions

Trigonometric functions, such as the cosine function, play a key role in modeling periodic phenomena. They allow the representation of oscillatory motion and are fundamental in problems involving angular measurement, especially in contexts such as circular or rotational movements.

Function Evaluation

Evaluating functions at specific input values involves substituting the given numbers into the function. This process is essential when determining the output of a model under particular conditions, and it illustrates how an abstract mathematical expression can yield concrete numerical results.

Algebraic Simplification

Algebraic simplification involves reducing expressions to more manageable forms. This is crucial when dealing with functions that combine multiple operations, such as trigonometric functions and radicals, so that the results can be correctly interpreted and computed.

Radical Expressions

Radical expressions, particularly square roots, appear in various modeling situations to represent non-linear relationships. Understanding how to simplify and manipulate these expressions is essential in obtaining an exact or approximate value when they are part of a function.

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