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Chapter 8: Problem 24

determine the quadrant in which \(\theta\) lies.. $$ \cos \theta>0, \tan \theta<0 $$

Short Answer

Expert verified
\(\theta\) lies in the fourth quadrant.

Step by step solution

01

Identify the quadrants in which cosine is positive

Cosine of an angle is positive in the first quadrant (0 < \(\theta\) < 90掳) and fourth quadrant (270 < \(\theta\) < 360掳). In other words, for an angle \(\theta\), \(\cos \theta > 0\) in the first and fourth quadrants.
02

Identify the quadrants in which tangent is negative

Tangent of an angle is negative in the second quadrant (90 < \(\theta\) < 180掳) and fourth quadrant (270 < \(\theta\) < 360掳). Therefore, for an angle \(\theta\), \(\tan \theta < 0\) in the second and fourth quadrants.
03

Find the quadrant where both conditions are satisfied

By combining the results from step 1 and step 2, it can be seen that both conditions, \(\cos \theta > 0\) and \(\tan \theta < 0\), are only satisfied in the fourth quadrant. Therefore, \(\theta\) lies in the fourth quadrant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrants
The concept of quadrants is essential when studying the trigonometric functions, especially in mathematics courses that cover topics like graph reading and function analysis. When we talk about quadrants, we are referring to the four sections of the Cartesian plane, which is divided by the x-axis and y-axis.
  • The first quadrant is where both x and y coordinates are positive.
  • The second quadrant is where x is negative and y is positive.
  • In the third quadrant, both x and y coordinates are negative.
  • The fourth quadrant features x as positive and y as negative.
Understanding which trigonometric functions are positive or negative in each quadrant is crucial for solving problems that involve angles and their trigonometric identities.
This setup helps sort out where an angle might lie based on given conditions, such as those involving cosine and tangent values.
Cosine Function
The cosine function, often denoted as \(\cos \theta \), measures the horizontal distance from the origin on a unit circle. Cosine deals with the adjacent side of a right triangle divided by the hypotenuse.
One important aspect of the cosine function is knowing where it is positive within the four quadrants:
  • The cosine function is positive in the first quadrant (0 < \( \theta \) < 90掳).
  • It is also positive in the fourth quadrant (270掳 < \( \theta \) < 360掳).
This is because both x-coordinates are positive here in terms of circle orientation. Recognizing where cosine is positive helps determine possible locations of angles on the coordinate plane, offering crucial data when solving angle-related problems.
Tangent Function
The tangent function (\( an \theta \)) signifies the ratio of the opposite side to the adjacent side in a right-angled triangle. It can be expressed using sine and cosine as \( an \theta = \frac{\sin \theta}{\cos \theta} \).
When evaluating the behavior of the tangent function across quadrants, consider:
  • Tangent is positive in the first (0掳 < \( \theta \) < 90掳) and the third quadrant (180掳 < \( \theta \) < 270掳).
  • It becomes negative in the second (90掳 < \( \theta \) < 180掳) and fourth quadrants (270掳 < \( \theta \) < 360掳).
This means if you have \( an \theta < 0 \), \( \theta \) could only reside in the second or fourth quadrant.
Combining this knowledge with the characteristics of other trigonometric functions helps pinpoint the exact quadrant for given angular conditions.

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Most popular questions from this chapter

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