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Function evaluation involves plugging specific values into a given function to compute its value. In our exercise, we have the function:\[f(r, \theta) = 2r(r \tan \theta - \sin 2\theta)\]To evaluate this function for particular arguments like \( f(3, \pi/4) \), you substitute the provided values of \( r \) and \( \theta \).

• First, substitute \( r = 3 \) and \( \theta = \pi/4 \) into the function.
• Next, compute the internal expressions \( r \tan \theta \) and \( \sin 2\theta \).
• After calculating these expressions, plug them back into the function equation.

This step-by-step evaluation reveals the computed values of the function for each provided pair of \( r \) and \( \theta \). The calculation is simple when values are correctly substituted and evaluated in order.

Converting angles is crucial when they're not initially within a manageable or familiar range, like between \( 0 \) and \( 2\pi \). In trigonometry, dealing with angles larger than \( 2\pi \) can be simplified by converting them into an equivalent angle within this range.The angle \( 9\pi/4 \) needs conversion. Here’s how:

• First, note that \( 2\pi \) is a full rotation. So subtract \( 2\pi \) from \( 9\pi/4 \).
• Calculate: \( 9\pi/4 - 2\pi = 9\pi/4 - 8\pi/4 = \pi/4 \).
• This means that \( 9\pi/4 \) is equivalent to \( \pi/4 \) within one full rotation.

When angles are reduced to a more familiar range, trigonometric functions become easier to evaluate as they involve well-known values.

Tangent and sine are fundamental trigonometric functions that relate angles to ratios of sides in right triangles.**Tangent Function**:The tangent of an angle \( \theta \) in a right triangle can be defined as the ratio of the opposite side to the adjacent side. For special angles like \( \pi/4 \), this function simplifies the expression because \( \tan(\pi/4) = 1 \), which makes calculations straightforward.**Sine Function**:The sine function measures the ratio of the opposite side to the hypotenuse of a right triangle. For example, when dealing with the double angle, \( \sin 2\theta \), a trigonometric identity is often used:\[\sin 2\theta = 2 \sin \theta \cos \theta\]For \( \pi/4 \), this simplifies since \( \sin(\pi/2) = 1 \), an easily memorizable and key value used in calculations.Understanding these functions makes tackling problems involving trigonometric evaluations much smoother and intuitive.