Problem 82 - Precalculus

Question

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by \(t\) lies. $$ \sin (t) > 0 \text { and } \cos (t) < 0 $$

Step-by-Step Solution

Answer

The point is in Quadrant II.

Step 1: Understand the Quadrants

In the coordinate plane, there are four quadrants. The sign of sine and cosine functions changes based on the quadrant: - In Quadrant I, \( \sin(t) > 0 \) and \( \cos(t) > 0 \)- In Quadrant II, \( \sin(t) > 0 \) and \( \cos(t) < 0 \)- In Quadrant III, \( \sin(t) < 0 \) and \( \cos(t) < 0 \)- In Quadrant IV, \( \sin(t) < 0 \) and \( \cos(t) > 0\).

Step 2: Identify the Correct Quadrant

We are given that \( \sin(t) > 0 \) and \( \cos(t) < 0 \). According to the quadrant characteristics in Step 1, these conditions match the signs in Quadrant II.

Step 3: Conclusion: Determine the Result

With \( \sin(t) > 0 \) and \( \cos(t) < 0 \), the point determined by \( t \) is in Quadrant II.

Key Concepts

Understanding Quadrants

In the world of trigonometry, understanding the coordinate plane is key. The plane is divided into four sections, known as quadrants. Each quadrant is defined by the positive or negative nature of the x and y coordinates.

Here's a quick guide to help you remember:

• **Quadrant I:** Both x and y coordinates are positive. Therefore, functions like the sine and cosine of an angle are both positive.
• **Quadrant II:** X coordinates are negative, while y coordinates are positive. As a result, the sine function is positive, but the cosine function is negative.
• **Quadrant III:** Both x and y coordinates are negative. Accordingly, both sine and cosine functions are negative.
• **Quadrant IV:** X coordinates are positive, while y coordinates are negative. Thus, the sine function is negative, and the cosine function is positive.

In the exercise, you determined the quadrant based on the signs of sine and cosine. Understanding the quadrant map is crucial for identifying where angles lie on the coordinate plane.

Sine Function

The sine function is a trigonometric function that emerges from the relation between the angles and sides of a right triangle. Specifically, it is the ratio of the opposite side to the hypotenuse:

• The formula for sine is given as \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \).

The sine function is central in understanding oscillations, waveforms, and circular motion.

It helps determine vertical positioning on the unit circle, where coordinates

• In Quadrant I, \( \sin(t) > 0 \)
• In Quadrant II, \( \sin(t) > 0 \)
• In Quadrant III, \( \sin(t) < 0 \)
• In Quadrant IV, \( \sin(t) < 0 \)

In our setting, since \( \sin(t) > 0 \), you're looking at either Quadrant I or II. This highlights its significance in determining which part of a circle or line an angle resides.

Cosine Function

The cosine function is another crucial trigonometric function, closely related to the sine function. It is defined as the ratio of the adjacent side over the hypotenuse in a right triangle:

• The formula for cosine is \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

Cosine describes horizontal positioning on the unit circle.

• In Quadrant I, \( \cos(t) > 0 \)
• In Quadrant II, \( \cos(t) < 0 \)
• In Quadrant III, \( \cos(t) < 0 \)
• In Quadrant IV, \( \cos(t) > 0 \)

In our exercise, given that \( \cos(t) < 0 \), Quadrant II becomes the focus. When combined with \( \sin(t) > 0 \), this solidly places the angle in Quadrant II, emphasizing the importance of understanding each function's sign.