Problem 5 - Contemporary Precalculus

Question

In Exercises \(1-6,\) determine the quadrant containing the termi. nal side of an angle of t radians in standard position under the given conditions. $$\sec t>0 \quad \text { and } \quad \cot t<0$$

Step-by-Step Solution

Answer

Answer: The terminal side of the angle t is located in the fourth quadrant.

Step 1: Understand the trigonometric functions and their signs

The secant (sec) function is the reciprocal of the cosine (cos) function, and the cotangent (cot) function is the reciprocal of the tangent (tan) function. To determine the quadrant where the terminal side of the angle is located, we should analyze the signs of the trigonometric functions in each quadrant. The cosine function is positive in the first (I) and the fourth (IV) quadrant, and negative in the second (II) and the third (III) quadrant. The tangent function is positive in the first and third quadrants, and negative in the second and fourth quadrants. Since sec(t) > 0 and cot(t) < 0, we will look for the quadrant where cosine is positive and tangent is negative.

Step 2: Determine the quadrant based on the signs of trigonometric functions

From the information above, we have: - Cosine is positive in quadrants I and IV, - Tangent is negative in quadrants II and IV. Now, to determine the quadrant where both conditions are met, we should look for the common quadrant in both lists. We find that the cosine is positive and the tangent is negative in the fourth quadrant. Therefore, the terminal side of the angle t is located in the fourth quadrant.

Key Concepts

Quadrant Analysis

Understanding which quadrant an angle resides in based on its trigonometric properties is crucial in trigonometry. The coordinate plane is divided into four quadrants: I, II, III, and IV. Each quadrant has unique sign rules for trigonometric functions.

For example:

• Quadrant I: All trigonometric functions have positive values.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

By identifying these sign rules, we can determine the quadrant of the terminal side of an angle under given conditions, such as \(\sec t > 0\) and \(\cot t < 0\). This tells us that the angle t falls into a specific section of the circle, recognizing its attributes from these trigonometric sign behaviors.

Secant Function

The secant function, denoted as \(\sec(t)\), is the reciprocal of the cosine function. That means \(\sec(t) = \frac{1}{\cos(t)}\). Since it's directly related to the cosine function, its behavior follows that of cosine.

Key properties of the secant function include:

• When cosine is positive, secant is positive.
• In quadrants I and IV, cosine and hence secant are positive.
• The secant function is undefined where cosine is zero (at \(p\),\(\frac{3\pi}{2}\), etc.).

Understanding the secant function helps us determine angles and their locations in different quadrants based on positive or negative signs.

Cotangent Function

Cotangent, represented as \(\cot(t)\), is the reciprocal of the tangent function, so \(\cot(t) = \frac{1}{\tan(t)}\). This function describes the ratio of the adjacent side to the opposite side in a right triangle.

In the unit circle:

• Cotangent is positive where tangent is positive (Quadrants I and III).
• Cotangent is negative where tangent is negative (Quadrants II and IV).
• This is useful when solving problems needing the cotangent sign, like \(\cot t < 0\). Knowing this helps us confirm that the angle lies in a quadrant where the tangent is negative, assisting us to predict the angle’s position accurately.

Trigonometric Sign Rules

The signs of trigonometric functions in each quadrant help in solving various mathematical problems. Remember, these rules define how sine, cosine, and tangent—and their reciprocals—behave based on the quadrant location.

Here’s a quick reminder:

• First Quadrant: All functions are positive.
• Second Quadrant: Sine is positive; others are negative.
• Third Quadrant: Tangent is positive; others are negative.
• Fourth Quadrant: Cosine is positive; others are negative.

These rules are not only tools for determining an angle’s position but are also essential for simplifying expressions and solving equations. Recognizing the quadrant will tell us about the sign of the function, aiding in accurate calculations.