Problem 96 - A Graphical Approach to Precalculus with Limits

Question

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\tan s<0, \sin s>0$$

Step-by-Step Solution

Answer

The point lies in the second quadrant.

Step 1: Understanding Quadrants

The coordinate plane is divided into four quadrants. In the first quadrant, both sine and tangent are positive. In the second quadrant, sine is positive, but tangent is negative. In the third quadrant, both sine and tangent are negative. In the fourth quadrant, sine is negative, and tangent is positive.

Step 2: Analyzing Given Conditions

We are given that \( \tan s < 0 \) and \( \sin s > 0 \). This means that the tangent is negative and sine is positive.

Step 3: Identifying the Suitable Quadrant

Based on the properties of the quadrants, we need a quadrant where sine is positive, but tangent is negative. Referring to our understanding from Step 1, the second quadrant is the only quadrant where this condition is met (sine positive, tangent negative).

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions. In a right-angled triangle, it is defined as the ratio of the length of the opposite side to the hypotenuse. Mathematically, this is expressed as:\[\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\]Sine is key to understanding the properties of angles and their respective quadrants on the unit circle. It provides an insight into the vertical component associated with an angle. The sine function graphs as a smooth, periodic wave stretching from -1 to 1. This reflects its cyclical nature and highlights when it is positive or negative depending on the quadrant. To understand how sine behaves within these quadrants:

• First Quadrant: Sine is positive.
• Second Quadrant: Sine remains positive.
• Third Quadrant: Sine becomes negative.
• Fourth Quadrant: Sine is negative again.

The positivity of sine in the second quadrant is crucial for solving problems like the one presented.

Tangent Function

The tangent function, another fundamental trigonometric function, is defined as the ratio of the sine to the cosine of an angle. This gives:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]The tangent function represents the slope of the line formed by the angle in standard position on the coordinate plane. It oscillates between negative and positive infinity and tends to have a more dramatic increase and decrease.Within different quadrants, the sign of tangent is determined by both sine and cosine:

• First Quadrant: Tangent is positive, as both sine and cosine are positive.
• Second Quadrant: Tangent is negative since sine is positive but cosine is negative.
• Third Quadrant: Tangent is positive again, because sine and cosine are both negative.
• Fourth Quadrant: Tangent is negative as sine is negative and cosine is positive.

This behavior is pivotal in finding the correct quadrant for angles with conditions like \(\tan s < 0\).

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis) intersecting at a central point known as the origin. This plane is essential for understanding how angles and trigonometric functions, like sine and tangent, operate across different quadrants. The plane is divided into four quadrants:

• First Quadrant: Both x and y coordinates are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

Each quadrant offers unique behaviors for trigonometric functions, impacting their positivity or negativity. This knowledge is crucial for deciphering exercises like identifying which quadrant aligns with certain trigonometric conditions.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and their measurements. The main trigonometric functions—sine, cosine, and tangent—are used to understand and describe the properties of angles and the triangles associated with them. These functions are periodic and exhibit patterns that repeat over a cycle.Apart from the primary functions, we have related functions like secant, cosecant, and cotangent. These complement and interact with the basic functions to provide more detailed insight into trigonometric problems.

• Sine and cosine oscillate between -1 and 1.
• Tangent, due to its divisive nature, fluctuates between negative and positive infinity.

This interplay helps solve complex trigonometric problems by determining the quadrant where specific conditions hold true, such as in our exercise with \(\tan s < 0, \sin s > 0\). Knowing the signs of these functions in each quadrant is essential for such determinations.