Problem 1 - Precalculus : Building Concepts and Connections

Question

For what value(s) of \(\theta\) in \([0,2 \pi]\) does \(\sin \theta\) reach a maximum value?

Step-by-Step Solution

Answer

The sine function reaches its maximum value for \( \theta = \frac{\pi}{2} \) in the interval [0,2π].

Step 1: Recognize the period of the sine function

The sine function has a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \) radians.

Step 2: Identify the maximum value of sine

The sine function reaches its maximum value of 1 at \( \theta = \frac{\pi}{2} \) within each period.

Step 3: Determine \( \theta \) in [0,2π] where sin reaches its max

Considering the restriction to the interval [0,2π], the value of \( \theta \) where \( \sin \theta \) reaches its maximum value is at \( \theta = \frac{\pi}{2} \), in radians.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry that relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. More generally, this concept extends to the unit circle, where any angle corresponds to a point on the circle, and the y-coordinate of that point is the sine of the angle.

Some key properties of the sine function include:

• The range of the sine function is from -1 to 1.
• It is an odd function, meaning even when the input sign is changed, the output sign also changes: \(\sin(-\theta) = -\sin(\theta)\).
• It is periodic with a specific repeating pattern.

Understanding these properties helps in analyzing the behavior of the sine function over different angles. The maximum value reached by sine, as with other trigonometric functions, is crucial for solving problems involving waves, simple harmonic motion, and more.

Periodic Functions

A periodic function is one that repeats its values at regular intervals. In trigonometry, sine is a classic example of a periodic function.

The sine function repeats itself every \(2\pi\) radians, which defines its period. This means:

• If you take any angle \(\theta\) and add \(2\pi\), the value of \(\sin\theta\) is the same.
• Mathematically, \(\sin(\theta + 2\pi n) = \sin\theta\), for any integer \(n\).
• This property makes sine useful in modeling cyclical phenomena like sound waves and tides.

Understanding periodicity allows us to predict behavior beyond a single range, making it a fundamental concept in analyzing trigonometric functions.

Radians

Radians are the standard unit of angular measure used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians connect angle measure to the radius of a circle.

Key points about radians include:

• One complete revolution around the circle is \(2\pidependent radians \), as the circle's circumference is \(2\pi r\) where \(r\) is the radius.
• Counterparts in degrees, 1 radian is equivalent to approximately 57.2958 degrees.
• Using radians simplifies calculus and other advanced areas in math.

When working with trigonometric functions, the use of radians allows for the smooth application of mathematical operations and integration with other mathematical principles. This makes radians an important tool for effectively analyzing trigonometric concepts, including solving problems like finding the maximum value of the sine function.