Problem 94 - 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. $$\tan s>0, \cos s<0$$

Step-by-Step Solution

Answer

The point lies in the 3rd quadrant.

Step 1: Understand the Tangent Condition

The condition \(\tan s > 0\) indicates that the tangent of angle \(s\) must be positive. Tangent is positive in the 1st and 3rd quadrants because \(\tan = \frac{\sin}{\cos}\) and both \(\sin\) and \(\cos\) are either both positive or both negative. Hence, \(\tan s > 0\) in these quadrants.

Step 2: Understand the Cosine Condition

The condition \(\cos s < 0\) states that the cosine of angle \(s\) must be negative. Cosine is negative in the 2nd and 3rd quadrants because the x-values (horizontal component in the unit circle) are negative there.

Step 3: Combine the Conditions

Combine the conditions to find the quadrant where both \(\tan s > 0\) and \(\cos s < 0\) are satisfied. From Step 1, \(\tan s > 0\) in the 1st and 3rd quadrants. From Step 2, \(\cos s < 0\) in the 2nd and 3rd quadrants. The quadrant that satisfies both conditions is the 3rd quadrant.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a visualization tool for understanding angles and their respective trigonometric values. It's a circle with a radius of 1, centered at the origin of a coordinate plane.

The unit circle helps you visualize the positions of angles and their associated sine, cosine, and tangent values, which derive from the circle's geometry. Here's how it works:

• Each point on the unit circle corresponds to an angle. The angle is typically measured from the positive x-axis.
• For any given angle, its coordinates \( (x, y) \) on the unit circle are \( (\cos s, \sin s) \). This means that the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine.
• Since the radius is 1, the circle’s equation is \( x^2 + y^2 = 1 \).

Understanding the unit circle is crucial for determining which trigonometric function is positive or negative, based on the angle's position in the different quadrants. Each quadrant on the unit circle indicates which trigonometric functions are positive or negative, aiding in solving problems related to conditions like \( \tan s > 0 \) or \( \cos s < 0 \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value in their domain. These identities are vital in simplifying trigonometric expressions and solving equations. Some of the most important identities include:

• Pythagorean identity: \( \sin^2 s + \cos^2 s = 1 \)
• Reciprocal identities: like \( \sec s = \frac{1}{\cos s} \)
• Ratio identities: \( \tan s = \frac{\sin s}{\cos s} \)
• Co-function identities: \( \sin(90^\circ - s) = \cos(s) \)

When dealing with problems involving conditions like \tan s > 0\ and \cos s < 0\, these identities help us understand the relationships between different trigonometric functions. Knowing that \( \tan s = \frac{\sin s}{\cos s} \) and its behavior in different quadrants makes it easier to deduce which quadrants meet specific conditions.

Tangent Function

The tangent function, represented as \( \tan s \), plays a crucial role in trigonometry, often used in equations involving angles and triangles. Defined by the ratio of \( \sin s \) to \( \cos s \), it informs us about the slope of an angle corresponding to a point on the unit circle.

Key aspects of the tangent function include:

• It is undefined when \( \cos s = 0 \) because that results in division by zero. This occurs at odd multiples of \90^\circ\ or \tt \frac{\pi}{2} \, \( \frac{3\pi}{2}, \ldots\).
• Tangent is positive in the 1st and 3rd quadrants, where \( \sin \) and \( \cos \) share the same sign, whether positive or negative.
• In the 2nd and 4th quadrants, the tangent function is negative because \( \sin \) and \( \cos \) have opposite signs.

Understanding the behavior of the tangent function in different quadrants helps solve problems like determining the location of \( s \) in the unit circle, when specific trigonometric conditions are given. The tangent function's characteristics are essential for verifying which quadrant matches conditions such as \( \tan s > 0 \).