Problem 13 - Calculus Single Variable

Question

Calculate the given expression without using a calculator. \(\sin (19 \pi / 2)^{\cos (33 \pi)}\)

Step-by-Step Solution

Answer

The expression equals 1.

Step 1: Convert Angles

First, we need to simplify the angles in their respective trigonometric functions. The angle \(19\pi/2\) can be expressed in terms of a multiple of \(2\pi\). The angle \(33\pi\) for cosine is sufficient because cosine has a period of \(2\pi\).

Step 2: Simplify Sine Argument

For \(\sin (19\pi/2)\), we first reduce it within the interval \(-\pi/2\) to \(\pi/2\). Notice that \(19\pi/2 = 9\pi + \pi/2\) or \(4\cdot 2\pi + \pi/2\). Therefore, \(\sin(19\pi/2) = \sin(\pi/2) = 1\).

Step 3: Evaluate Cosine Expression

For \(\cos (33\pi)\), we simplify the angle: since \(33\pi = 16 \cdot 2\pi + \pi\), cosine at any odd multiple of \(\pi\) flips to \(-1\). Thus, \(\cos (33\pi) = \cos(\pi) = -1\).

Step 4: Combine to Solve Exponentiation

With \(\sin (19\pi/2) = 1\) and \(\cos (33\pi) = -1\), the expression becomes \(1^{-1}\). Since any non-zero number raised to the power of \(-1\) is its reciprocal, \(1\) raised to any power remains \(1\). Thus, the result of \(\sin (19\pi/2)^{\cos (33\pi)} = 1\).

Key Concepts

Angle Reduction

Angle reduction is useful in simplifying trigonometric expressions. It involves reducing angles to their smallest possible equivalent within a given interval, typically (-\pi, \pi]\ or (-\pi/2, \pi/2]\, because the sine and cosine functions are periodic. The function of angle reduction is crucial for calculating expressions without a calculator.

• For \(19\pi/2\), this angle can be simplified by identifying multiples of \(2\pi\), which represents a full circle (360 degrees). In this case, \(19\pi/2 = 9\pi + \pi/2 = 4\cdot 2\pi + \pi/2 \).
• This simplification allows us to express \(\sin(19\pi/2) \)} as \(\sin(\pi/2)\).

Using these reductions ensures the calculation is straightforward, as expressed angles are equivalent in trigonometric functions.

Sine Function

The sine function is a fundamental part of trigonometry. It represents the y-coordinate of a point on the unit circle, which correlates to an angle's height above the horizontal axis. The sine function has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.

• In the exercise, \(\sin(19\pi/2)\) reduces to \(\sin(\pi/2)\), which equals 1.
• This is because \(\sin(\pi/2)\) corresponds to the topmost point on the unit circle, where the y-coordinate is 1.

Knowing how to interpret and simplify angles within the sine function helps streamline calculations, rendering the process more intuitive and straightforward.

Cosine Function

The cosine function, like sine, measures a point on the unit circle. However, it represents the x-coordinate, correlating to an angle's horizontal distance from the origin. The cosine function also has a period of \(2\pi\).

• For \(\cos(33\pi)\), dealing with an odd multiple of \(\pi\) (i.e., 16 multiples of \(2\pi\) plus an extra \(\pi\)), you utilize the symmetry property: \(\cos(\pi) = -1\).
• This crucial property simplifies the evaluation of cosine at large angles, accounting for its periodic nature.

Mastering such reductions and transformations in expressions involving cosine leads to quick, efficient problem-solving in trigonometry.