Problem 49 - Algebra and Trigonometry

Question

From the information given, 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 terminal point determined by \(t\) lies in the second quadrant.

Step 1: Understanding Sine and Cosine

The sine function, \( \sin t \), is positive in the first and second quadrants. The cosine function, \( \cos t \), is negative in the second and third quadrants.

Step 2: Identifying the Common Quadrant

Combine the information. We know \( \sin t > 0 \) (first and second quadrants) and \( \cos t < 0 \) (second and third quadrants). The only quadrant where both conditions are true simultaneously is the second quadrant.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, represented by the symbol \( \sin \). It measures the y-coordinate of a point on the unit circle, which is a circle with a radius of one centered at the origin of a coordinate system. The value of \( \sin t \) corresponds to how high above or below the x-axis a point lies when you move counterclockwise from the positive x-axis by an angle \( t \).
Here's an easy way to remember its values in relation to the quadrants:

• In Quadrant I, both sine and cosine are positive, so \( \sin t > 0 \).
• In Quadrant II, sine remains positive while cosine becomes negative, \( \sin t > 0 \).
• In Quadrant III and IV, the sine function becomes negative because points are either directly below or towards the negative side of the x-axis.

Understanding where \( \sin t \) is positive is crucial for solving many trigonometric problems, such as determining the quadrant of an angle.

Cosine Function

The cosine function, denoted by \( \cos \), is another key function in trigonometry. It represents the x-coordinate on the unit circle. Essentially, \( \cos t \) tells us how far left or right a point on the unit circle is from the y-axis after rotating an angle \( t \).
Here's how the cosine function behaves across the quadrants:

• In Quadrant I, cosine is positive because the points are to the right of the y-axis, \( \cos t > 0 \).
• In Quadrant II, cosine becomes negative because points are now to the left of the y-axis, \( \cos t < 0 \).
• In Quadrant III, cosine remains negative as the points continue to be on the left.
• In Quadrant IV, cosine turns positive again.

By analyzing the cosine values, one can easily identify in which quadrant a particular angle \( t \) lies when given additional conditions.

Quadrant Analysis

Quadrant Analysis helps us determine the location of an angle or point on the coordinate plane based on trigonometric values. The coordinate plane is divided into four regions, or quadrants:

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

Our task is to determine the correct quadrant given specific sine and cosine conditions.
In the provided exercise, the conditions \( \sin t > 0 \) and \( \cos t < 0 \) were given. Using quadrant analysis, we find:- \( \sin t > 0 \) limits the possibilities to Quadrants I and II.- \( \cos t < 0 \) restricts it further to Quadrants II and III.By intersecting these conditions, only Quadrant II satisfies both, as it is the only quadrant where sine is positive and cosine is negative. This analytic approach simplifies the process of determining the correct quadrant efficiently.