Problem 51 - Algebra and Trigonometry

Question

From the information given, find the quadrant in which the terminal point determined by \(t\) lies. \(\sin t>0\) and \(\cos t<0\)

Step-by-Step Solution

Answer

Quadrant II.

Step 1: Understand the Quadrants

The coordinate plane is divided into four quadrants. In these quadrants: 1. Quadrant I: \( ext{Both } \ \ ext{sin}t \ ext{ and cos}t \ ext{ are positive.}\)2. Quadrant II: \( ext{sin}t > 0 \ ext{ and cos}t < 0.}\)3. Quadrant III: \( ext{Both } \ \ ext{sin}t \ ext{ and cos}t \ ext{ are negative.}\)4. Quadrant IV: \(\ \ \ ext{sin}t < 0 \ ext{ and cos}t > 0.}\)

Step 2: Analyze the Conditions Given

We are given that \(\ \ \ ext{sin}t > 0\ \ \ \) and \(\ \ \ ext{cos}t < 0.\ \ \) Based on our understanding of quadrants, we know \(\ \ \ ext{sin}t > 0\ \ \ \) occurs in Quadrants I and II, and \(\ \ \ ext{cos}t < 0\ \) occurs in Quadrants II and III. The only quadrant where both these conditions are met is Quadrant II.

Key Concepts

Sine Function

The sine function, often referred to as `sin`, is crucial in trigonometry. It relates an angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. In the context of the coordinate plane, the sine of an angle reflects the y-coordinate of a point on the unit circle. This is because the unit circle has a radius of 1, making the hypotenuse always equal to 1.

The sine function has some specific properties:

• Its value ranges between -1 and 1.
• It is positive in Quadrants I and II of the coordinate plane.
• It is negative in Quadrants III and IV.

When analyzing the given exercise where \( \sin t > 0 \), this information helps us conclude that the angle or terminal point must be in either Quadrant I or Quadrant II, where the sine function is positive.

Cosine Function

The cosine function, frequently abbreviated as `cos`, is another fundamental trigonometric function. It compares an angle to the ratio of the adjacent side to the hypotenuse in a right triangle.

On the unit circle, cosine corresponds to the x-coordinate of a point, sharing a connection with the circle's horizontal axis. Similar to sine, the cosine's values are also bounded between -1 and 1.

• Cosine is positive in Quadrants I and IV.
• It remains negative in Quadrants II and III.

For the provided situation, \( \cos t < 0 \), the cosine indicates that the terminal point must be in either Quadrant II or Quadrant III. This insight, combined with the sine function's condition, helps to pinpoint the specific quadrant needed for the solution.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by perpendicular axes—namely the x-axis (horizontal) and y-axis (vertical). It is divided into four sections, termed quadrants, which are essential for understanding trigonometric functions.

Each quadrant in the coordinate plane has its distinct characteristics regarding the sign of sine and cosine.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine become negative.
• Quadrant IV: Sine is negative, but cosine is positive.

When tasked with identifying the quadrant for a particular terminal point, knowledge of these characteristics is pivotal. In the exercise, recognizing that \( \sin t > 0 \/ \cos t < 0 \) directs us to Quadrant II, where those conditions align perfectly, thereby solving the problem.