Problem 33 - Algebra and Trigonometry

Question

33–36 Find the quadrant in which \(\theta\) lies from the information given. \(\sin \theta<0 \quad\) and \(\quad \cos \theta<0\)

Step-by-Step Solution

Answer

\( \theta \) lies in Quadrant III.

Step 1: Identify Sine and Cosine Signs

First, determine the sign of the sine and cosine in each quadrant.- Quadrant I: \( \sin \theta > 0 \), \( \cos \theta > 0 \)- Quadrant II: \( \sin \theta > 0 \), \( \cos \theta < 0 \)- Quadrant III: \( \sin \theta < 0 \), \( \cos \theta < 0 \)- Quadrant IV: \( \sin \theta < 0 \), \( \cos \theta > 0 \)

Step 2: Apply Given Conditions

The conditions provided are \( \sin \theta < 0 \) and \( \cos \theta < 0 \). Review each quadrant:- Quadrant I: Doesn't satisfy conditions (none are negative).- Quadrant II: Doesn't satisfy conditions (sine is positive).- Quadrant III: Satisfies both conditions (both sine and cosine are negative).- Quadrant IV: Doesn't satisfy conditions (cosine is positive).

Step 3: Conclusion

From the analysis in Step 2, we concluded that only Quadrant III satisfies \( \sin \theta < 0 \) and \( \cos \theta < 0 \). Therefore, \( \theta \) lies in Quadrant III.

Key Concepts

Understanding Quadrants

In trigonometry, the coordinate plane is divided into four sections known as quadrants. Each quadrant represents a specific range of angle measurements and characteristics for trigonometric functions. Knowing which quadrant an angle lies in is crucial for determining the signs of the sine, cosine, and tangent of that angle.

Here is a simple breakdown of the quadrants:

• Quadrant I: Angles between 0 and 90 degrees. Here both sine and cosine are positive.
• Quadrant II: Angles between 90 and 180 degrees. In this quadrant, sine is positive while cosine is negative.
• Quadrant III: Angles between 180 and 270 degrees. Both sine and cosine have negative values here, which is useful for certain types of calculations, especially when solving problems similar to the exercise given.
• Quadrant IV: Covers angles from 270 to 360 degrees, where sine is negative and cosine is positive.

It's essential to memorize these properties as it helps in solving trigonometry problems quickly.

Signs of Sine and Cosine

The signs of the sine and cosine functions are pivotal in identifying which quadrant an angle belongs to. The behavior of these functions depends on the quadrant in which the angle is located.

Here's a quick guide on the signs of the sine and cosine for each quadrant:

• In Quadrant I: Both sine (\( heta > 0\)) and cosine (\( heta > 0\)) are positive, making calculations straightforward.
• Quadrant II: Sine remains positive (\( heta > 0\)) while cosine becomes negative (\( heta < 0\)). This change affects the results of calculations involving these functions.
• Quadrant III offers a twist, as both sine and cosine become negative (\( heta < 0\)), which may be useful in certain trigonometric solutions and inverses.
• In Quadrant IV: The sine function becomes negative (\( heta < 0\)) alongside a positive cosine (\( heta > 0\)).

Understanding these signs helps simplify the solution of any problem requiring identification of angle placement and value computation based on these trigonometric functions.

Solving Trigonometry Problems

Solving trigonometry problems often requires knowledge of the signs and properties of trigonometric functions across the quadrants. This knowledge informs the strategies you will use in determining the correct quadrant and evaluating functions.

Here’s a simple approach to tackle trigonometry problems like the exercise above:

• Begin by identifying the signs of trigonometric functions given in the problem. This narrows down potential quadrants quickly.
• Match these signs with the quadrant properties. This provides a direct approach to pinpointing the right quadrant, as seen when both \( heta\) and sine are negative, aligning with Quadrant III.
• Applying these observations reduces complicated calculations and helps verify the path to the solution swiftly.
• Practice with varying conditions to reinforce understanding and agility in identifying correct quadrants and figuring out trigonometric identities.

With practice and familiarity with these patterns, solving even complex trigonometry problems can become a straightforward task.