Problem 48 - Precalculus Mathematics for Calculus

Question

Find the sign of the expression if the terminal point determined by \(t\) is in the given quadrant. \(\tan t\) sec \(t, \quad\) Quadrant IV

Step-by-Step Solution

Answer

In Quadrant IV, \( \tan t \sec t \) is negative.

Step 1: Determine Sign of Trigonometric Functions

In Quadrant IV, the cosine of an angle is positive, while the tangent is negative. This information is essential for determining the sign of the expression. Specifically, for angles in Quadrant IV, \( \tan t < 0 \) and \( \sec t > 0 \).

Step 2: Recall Trigonometric Identities

Remember that the secant function is the reciprocal of the cosine function, meaning \( \sec t = \frac{1}{\cos t} \). Therefore, \( \sec t \) retains the same sign as \( \cos t \). Since \( \cos t > 0 \) in Quadrant IV, \( \sec t > 0 \).

Step 3: Analyze the Expression

The expression is \( \tan t \sec t \). Given that \( \tan t < 0 \) and \( \sec t > 0 \), the product of these two terms will be negative because the multiplication of a negative and a positive number results in a negative number.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold true for any angle. Understanding these identities helps simplify expressions and solve trigonometric equations effectively.

• One important identity is that the secant (\( \sec t \)) function is the reciprocal of the cosine (\( \cos t \)) function, mathematically represented as \( \sec t = \frac{1}{\cos t} \).
• The tangent (\( \tan t \)) function can be expressed in terms of sine (\( \sin t \)) and cosine as \( \tan t = \frac{\sin t}{\cos t} \).

These identities are beneficial when determining the sign of trigonometric functions depending on the quadrant. Since \( \sec t \) is the reciprocal of \( \cos t \), they share the same sign. Being familiar with these identities makes identifying the sign of expressions such as \( \tan t \sec t \) straightforward when you know in which quadrant the angle lies.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four sections known as quadrants. Each quadrant has unique characteristics that affect the signs of trigonometric functions:

• Quadrant I (0° to 90°): All trigonometric functions are positive.
• Quadrant II (90° to 180°): Sine is positive, cosine and tangent are negative.
• Quadrant III (180° to 270°): Tangent is positive, sine and cosine are negative.
• Quadrant IV (270° to 360°): Cosine is positive, sine and tangent are negative.

Knowing the quadrant in which an angle terminates is crucial for accurately determining the sign of trigonometric functions. In the given exercise, the terminal point lies in Quadrant IV where \( \cos t > 0 \) and \( \tan t < 0 \). This information helps in figuring out that the expression \( \tan t \sec t \) will result in a negative sign as the product of a positive and a negative number is negative.

Sign of Trigonometric Functions

To correctly determine the sign of trigonometric expressions, we rely on the signs of individual trigonometric functions within their respective quadrants. Each function—sine, cosine, tangent—has a different behavior as the angle traverses through the quadrants.
When we discuss the sign of a trigonometric expression, we need to consider:

• The sign of each individual function in the expression.
• The impact of any identities that may transform the functions.

In Quadrant IV, \( \tan t \) is negative because tangent is negative there. However, \( \sec t = \frac{1}{\cos t} \) remains positive since cosine is positive in this quadrant. Hence, when multiplying \( \tan t \) with \( \sec t \), the resulting value is negative because one positive and one negative number multiplied gives a negative result.