Problem 101 - Precalculus

Question

Determine whether each statement is true or false. \(\sec ^{2} \theta-1\) can be negative for some value of \(\theta\)

Step-by-Step Solution

Answer

The statement is false; \( \sec^2 \theta - 1 \) cannot be negative for any \( \theta \).

Step 1: Recall the identity

Recall the trigonometric identity: \( \sec^2 \theta - 1 = \tan^2 \theta \). This identity is derived from the Pythagorean identity for sine and cosine.

Step 2: Understand the expression

The expression \( \sec^2 \theta - 1 \) is equivalent to \( \tan^2 \theta \) based on the identity. We need to determine if \( \tan^2 \theta \) can be negative.

Step 3: Evaluate \( \tan^2 \theta \)

Since \( \tan^2 \theta = (\tan \theta)^2 \), this is always non-negative because the square of any real number (positive, negative, or zero) is non-negative.

Step 4: Conclusion

Since \( \tan^2 \theta \) cannot be negative for any real number \( \theta \), \( \sec^2 \theta - 1 \) cannot be negative for any value of \( \theta \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), plays an important role in trigonometry. It is defined as the reciprocal of the cosine function. Hence, we write it as \( \sec \theta = \frac{1}{\cos \theta} \).
Understanding this relationship is essential when dealing with trigonometric identities, such as the Pythagorean identity.

• The secant function is undefined wherever the cosine function is zero because division by zero is undefined in mathematics.
• Common angles where this occurs are \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \).
• The range of the secant function excludes the interval from -1 to 1, focusing instead on values greater than or equal to 1 or less than or equal to -1.

The given exercise uses the secant function to explore its squared value minus one, which is tightly related to the tangent function.

Pythagorean Identity

The Pythagorean identity in trigonometry connects the sine and cosine functions in a beautiful way. Written as \( \sin^2 \theta + \cos^2 \theta = 1 \), it creates a harmonious relationship between these two primary trigonometric functions.
From this fundamental identity, other related identities can be derived:

• By dividing the entire equation by \( \cos^2 \theta \), we obtain \( \tan^2 \theta + 1 = \sec^2 \theta \).
• Alternatively, dividing by \( \sin^2 \theta \) furnishes another identity: \( 1 + \cot^2 \theta = \csc^2 \theta \).

In the exercise, the identity \( \sec^2 \theta - 1 = \tan^2 \theta \) is derived, showcasing the power of the original Pythagorean identity in transforming and simplifying complex expressions.

Tangent Function

The tangent function, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), holds a critical place in trigonometry, often representing the slope of an angle in a right triangle. Its properties significantly influence how trigonometric identities are analyzed and understood.

• The tangent function is periodic, with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.
• The function is undefined where the cosine, its denominator, equals zero. This occurs at \( \theta = \frac{\pi}{2} + n\pi \).
• Importantly, \( \tan^2 \theta \) can never be negative because squaring any real number results in a non-negative outcome.

In the problem, you're asked to determine whether \( \sec^2 \theta - 1 \) can be negative by understanding that it equates to \( \tan^2 \theta \). Recognizing that \( \tan^2 \theta \) yields non-negative values clarifies that the expression cannot be negative.