Problem 139 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. $$\sin \theta<\tan \theta \text { in Quadrant I }$$

Step-by-Step Solution

Answer

The statement that \(\sin \theta < \tan \theta\) in Quadrant I is generally true, except when \(\theta = \frac{\pi}{4}\), at which point \(\sin \theta = \tan \theta = \frac{1}{\sqrt{2}}\).

Step 1: Understand the statement

The statement represents an inequality between the sine and tangent of an angle \(\theta\) in the first quadrant. In the first quadrant, both sine and tangent are positive, as \(\theta\) ranges between 0 and \(\frac{\pi}{2}\) radians.

Step 2: Compare Sine and Tangent

In the first quadrant, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), whereas \(\sin \theta = \sin \theta \times \frac{1}{1}\). If cos(\theta) is greater than 1, the value of \(\sin \theta\) is greater than \(\tan \theta\). However, in the first quadrant, \(\cos \theta\) lies between 0 and 1. Hence, \(\sin \theta < \tan \theta\) in the first quadrant is generally true.

Step 3: Identify the exception

Although the statement is mostly true, there is one exception. When \(\theta = \frac{\pi}{4}\), \(\sin \theta = \tan \theta = \frac{1}{\sqrt{2}}\). Therefore, at this specific point, \(\sin \theta\) is not less than \(\tan \theta\).

Key Concepts

Trigonometric Functions

Trigonometric functions are the foundation of trigonometry and are used to describe the relationship between the angles and sides of triangles, particularly right-angled triangles. The primary trigonometric functions include sine (\texttt{sin}), cosine (\texttt{cos}), and tangent (\texttt{tan}). Each function represents a ratio of sides of a right triangle in relation to an acute angle of the triangle.

For an angle \( \theta \) in a right-angled triangle, \( \sin \theta \) is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Meanwhile, \( \tan \theta \) is the ratio of the length of the side opposite to \( \theta \) to the length of the side adjacent to \( \theta \). Understanding these functions is essential as they are not only used within the realm of geometry but are also applied across various scientific and engineering disciplines.

First Quadrant

The first quadrant is one of the four divisions of the coordinate plane, where both x and y coordinates are positive. For the angle \( \theta \) in standard position, whose terminal side lies in the first quadrant, its measure would be greater than 0 but less than \( \frac{\pi}{2} \) radians (or 90 degrees).

In this quadrant, all primary trigonometric functions are positive. This implies that both \( \sin \theta \) and \( \cos \theta \) (and therefore \( \tan \theta \) as the ratio of these two) will yield positive values. Understanding this positivity is crucial when approaching problems involving trigonometric inequalities in the first quadrant.

Inequalities in Trigonometry

In trigonometry, inequalities help determine the range and relation among trigonometric functions for different angles. The inequality \( \sin \theta < \tan \theta \) expresses a comparative relationship between the sine and tangent of an angle \( \theta \).

However, when dealing with inequalities, it's vital to remember that the relationship can change depending on the quadrant and specific values of \( \theta \). As noted in the solution steps for our exercise, \( \sin \theta \) is typically less than \( \tan \theta \) in Quadrant I with an exception at \( \theta = \frac{\pi}{4} \), where they are equal. This concept helps in predicting and understanding the behavior of trigonometric functions across different ranges.

Identities and Properties of Trigonometry

The rich field of trigonometry is undergirded by several identities and properties that define the inherent connections between its functions. Some of the most common trigonometric identities involve expressions of \texttt{sin}, \texttt{cos}, and \texttt{tan} in forms that allow for simplification and transformation of trigonometric expressions.

An example is the Pythagorean identity, which shows that \( \sin^2 \theta + \cos^2 \theta = 1 \). It's also important to note properties like the periodicity of these functions, which indicate that they repeat their values in regular intervals. Furthermore, the reciprocal and ratio identities link \texttt{sin}, \texttt{cos}, and \texttt{tan} in a way that clarifies their interdependence. Recognizing these identities and properties equips students with the ability to manipulate and compare trigonometric functions effectively, which is particularly helpful when proving inequalities.