Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sin \theta>0, \tan \theta>0$$

Step-by-Step Solution

Answer

The angle \(\theta\) is in Quadrant I.

Step 1: Understand Sine Condition

The sine of an angle, \(\sin \theta\), is positive in Quadrants I and II. This means that \(\theta\) can lie in either of these two quadrants.

Step 2: Understand Tangent Condition

The tangent of an angle, \(\tan \theta\), is positive in Quadrants I and III. This is because tangent is the ratio of sine to cosine. Since both sine and cosine are either positive or negative in these quadrants, their division is positive.

Step 3: Combine Conditions

To satisfy both \(\sin \theta>0\) and \(\tan \theta>0\), \(\theta\) must lie in Quadrant I. This is the only quadrant where both conditions are true simultaneously.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the properties and relationships of angles. These functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), among others. One of their basic uses is to determine the relationship between angles and side lengths in right triangles.

When dealing with these functions, it's important to know that:

• The sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the sine to the cosine, meaning it is the opposite over adjacent side.

In trigonometry, angles can be positioned in what's known as the unit circle, which helps us to understand the sign and value of these functions in different quadrants. Each function has properties of being positive or negative depending on the quadrant in which the angle lies.

This foundational understanding helps when identifying which quadrant an angle is in based on the positivity or negativity of these fundamental trigonometric functions.

Sine Function

The sine function, represented as \(\sin\theta\), is one of the primary trigonometric functions used in mathematics. One of its main characteristics is to describe how a particular angle relates to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

When using the unit circle:

• The sine function is positive when \(\theta\) is in Quadrants I and II.
• In Quadrant III and IV, the sine function becomes negative.

In the case of the original exercise, we are told that \(\sin\theta > 0\). This indicates that our angle, \(\theta\), can only be in Quadrants I or II.

Understanding when the sine function is positive is crucial when you are tasked with determining quadrant locations, especially when combined with other trigonometric properties.

Tangent Function

The tangent function, noted as \(\tan\theta\), is essential when analyzing angle relationships since it combines both the sine and cosine functions. Essentially, the tangent of an angle is given by the formula \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), meaning it derives its value from the other two primary trigonometric functions.

The sign of the tangent function depends on the signs of \(\sin\) and \(\cos\):

• Tangent is positive when both sine and cosine have the same sign.
• Tangent is negative when sine and cosine have opposite signs.

Thus, \(\tan\theta\) is positive in Quadrants I and III because:

• In Quadrant I, both sine and cosine are positive.
• In Quadrant III, both sine and cosine are negative, making their ratio positive.

For our original exercise, \(\tan\theta > 0\) suggests \(\theta\) could lie in either Quadrant I or III. When combined with the sine condition, quadrants are narrowed down further.