Problem 45 - Basic Technical Mathematics with Calculus

Question

Explain why the given statements are true for an acute angle \(\theta\).\(\sin \theta\) is always between 0 and 1.

Step-by-Step Solution

Answer

\( \sin \theta \) for an acute angle is between 0 and 1 because \( \theta \) produces a positive y-coordinate in the unit circle, but less than the circle's radius.

Step 1: Understand the Sine Function

The sine function, represented as \( \sin \theta \), calculates the opposite side over the hypotenuse in a right-angled triangle. In a unit circle, \( \sin \theta \) corresponds to the y-coordinate of the point at an angle \( \theta \).

Step 2: Define the Range of an Acute Angle

An acute angle \( \theta \) is defined as an angle that is greater than 0 degrees and less than 90 degrees. This means \( 0^{\circ} < \theta < 90^{\circ} \).

Step 3: Analyze the Unit Circle Representation

In a unit circle, every angle \( \theta \) is formed using a radius of 1. For any acute angle, \( \sin \theta \) is the y-coordinate of the point on the circle, which lies between 0 and the maximum radius of the circle, which is 1.

Step 4: Conclude the Range of \( \sin \theta \)

Since \( \theta \) is an acute angle, the sine value, \( \sin \theta \), represents a positive y-coordinate in the unit circle. The value cannot be 0 (as 0 degrees represents the x-axis) and cannot reach 1, as that is the value for 90 degrees. Hence, 0 < \( \sin \theta \) < 1 for \( \theta \) acute.

Key Concepts

Sine Function

The sine function, one of the fundamental trigonometric functions, is central in studying triangles. When we refer to the sine of an angle \( \theta \), denoted as \( \sin \theta \), we're essentially looking at the relationship between two sides of a right-angled triangle: the side opposite the angle and the triangle's hypotenuse.
While in a triangle, the formula is expressed as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
This gives you an idea of how far up or down you are against the hypotenuse of the triangle.

Understanding this function is crucial as it allows you to determine the value of an angle or the lengths of a triangle's other sides when given certain measures.

• In practical applications, the sine function helps in determining heights and distances where direct measurement is challenging or impossible.
• It's also vital in other areas like signal processing and harmonics due to its periodic nature.

Acute Angles

An acute angle is an angle that measures more than 0 degrees but less than 90 degrees. These angles are commonly found in various shapes and geometrical constructs, making an understanding of them essential.
For example, all angles in an equilateral triangle are acute angles, each measuring 60 degrees.

A simple way to visualize acute angles is by comparing them to right angles, which are exactly 90 degrees. If it 'looks' smaller than a right angle, it's likely acute. Here's why they matter in our context:

• When dealing with right-angled triangles, acute angles help calculate the sine, since these angles determine the ratio of sides.
• They are useful in defining the types and properties of polygons and in calculating areas and shapes in various problems.

Unit Circle Representation

The unit circle is a powerful tool in trigonometry, providing a geometric representation of trigonometric functions.
It is a circle with a radius of 1, centered at the origin of a coordinate plane. For a given angle \( \theta \), the point on the unit circle can be represented as \((\cos \theta, \sin \theta)\), where \( \sin \theta \) aligns with the y-coordinate of the point.

This representation simplifies understanding trigonometric concepts.
Using the unit circle:

• The maximum value \( \sin \theta \) reaches is 1, occurring right at 90 degrees, the top of the circle.
• For any angle less than 90 degrees but more than 0 degrees (an acute angle), \( \sin \theta \) captures the height from the x-axis to the point on the circle.

Hence, for any acute angle, the sine value always stays between 0 and 1. This visual tool not only aids in visualizing how trigonometric functions work but also makes calculation more intuitive by
transforming abstract numerical exercises into clear geometric contexts.