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Chapter 4: Problem 22

State the quadrant in which \(\theta\) lies. $$\sec \theta>0 \text { and } \cot \theta<0$$

Short Answer

Expert verified
The given conditions are met in fourth quadrant.

Step by step solution

01

Understanding Secant

Recall that secant, or \(\sec \theta\), is positive in the first and fourth quadrants. This is because secant is the inverse of cosine, and cosine is positive in these quadrants.
02

Understanding Cotangent

Remember that cotangent, or \(\cot \theta\), is negative in the second and fourth quadrants. This is due to the fact that cotangent is the inverse of tangent, and tangent is negative in these quadrants.
03

Finding the Common Quadrant

Now you have the information to determine where \(\theta\) lies. Since both criteria indicate that \(\theta\) should be in the fourth quadrant, this is the answer.

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Key Concepts

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

Understanding Quadrants
In trigonometry, the coordinate plane is divided into four quadrants. These quadrants help us understand the signs of trigonometric functions.
  • Quadrant I: All trigonometric functions are positive.
  • Quadrant II: Only sine and cosecant are positive.
  • Quadrant III: Only tangent and cotangent are positive.
  • Quadrant IV: Only cosine and secant are positive.
Recognizing these properties is key to solving trigonometric problems by determining the sign of functions like \(\sec \theta\) and \(\cot \theta\). Different functions are positive or negative depending on their relationship to angles and reference axes.
Understanding these differences allows you to place \(\theta\) accurately within the correct quadrant on the coordinate plane.
Exploring Secant
Secant, denoted as \(\sec \theta\), is the reciprocal of cosine. So, \(\sec \theta = \frac{1}{\cos \theta}\). Because of this relationship, secant inherits the sign of cosine.
  • Positive in Quadrant I: Where cosine is positive.
  • Positive in Quadrant IV: Where cosine is also positive.
  • Negative in Quadrants II and III: Where cosine is negative.
Secant helps determine properties of angles through its reciprocal relationship with cosine. This makes it vital for angle calculations and analysis in trigonometry.
Proper understanding of secant allows for determining where certain angle values exist within different quadrants.
Deciphering Cotangent
Cotangent, denoted as \(\cot \theta\), is the reciprocal of tangent, so \(\cot \theta = \frac{1}{\tan \theta}\). As tangent is positive in certain quadrants, cotangent will be negative in others.
  • Positive in Quadrant III: Similar to tangent.
  • Positive in Quadrant I (though less common).
  • Negative in Quadrants II and IV: Where tangent is negative.
Understanding cotangent requires grasping how tangent behaves across the quadrants. This helps in identifying angles where specific trigonometric conditions occur.
Cotangent plays a crucial role in solving problems where angle determination involves reciprocal logic.
Determining Angles with Conditions
When given conditions like \(\sec \theta > 0\) and \(\cot \theta < 0\), angle determination involves analyzing which quadrant satisfies both.
1. Note that \(\sec \theta\) is positive in Quadrants I and IV.2. Recognize that \(\cot \theta\) is negative in Quadrants II and IV.3. The overlap is in Quadrant IV, where both conditions are met.By applying these conditions, we can conclude that \(\theta\) lies in Quadrant IV.
This method of determining angles ensures accuracy in placing angles based on trigonometric function signs.

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Most popular questions from this chapter

Find a model for simple harmonic motion satisfying the specified conditions. $$\begin{array}{cc}\text{Displacement \((t=0)\)} & \text{Amplitude} & \text{Period} \\ 0& 4 \mathrm{centimeters}& 2 \mathrm{seconds}\end{array}$$

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Find the length of the sides of a regular hexagon inscribed in a circle of radius 25 inches.
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