Problem 130 - A Graphical Approach to Precalculus with Limits

Question

Use fundamental identities to find each expression. Write \(\sin \theta\) in terms of \(\sec \theta\) if \(\theta\) is in quadrant I or II.

Step-by-Step Solution

Answer

\(\sin \theta = \sqrt{1 - \frac{1}{\sec^2 \theta}}\) for \(\theta\) in quadrant I or II.

Step 1: Identify the relevant identities

First, recall the fundamental trigonometric identities. We can use the identity for secant: \( \sec \theta = \frac{1}{\cos \theta} \). Additionally, recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 2: Express \(\cos^2 \theta\) using \(\sec \theta\)

Since \(\sec \theta = \frac{1}{\cos \theta}\), it implies \(\cos \theta = \frac{1}{\sec \theta}\). Consequently, \(\cos^2 \theta = \left( \frac{1}{\sec \theta} \right)^2 = \frac{1}{\sec^2 \theta}\).

Step 3: Use the Pythagorean identity to find \(\sin^2 \theta\)

Substitute \(\cos^2 \theta = \frac{1}{\sec^2 \theta}\) into the identity \(\sin^2 \theta + \cos^2 \theta = 1 \), to get \(\sin^2 \theta = 1 - \frac{1}{\sec^2 \theta}\).

Step 4: Solve for \(\sin \theta\)

Take the square root of both sides to solve for \(\sin \theta\), giving \(\sin \theta = \pm \sqrt{1 - \frac{1}{\sec^2 \theta}}\).

Step 5: Determine the correct sign for \(\sin \theta\) based on the quadrant

Since \(\theta\) is in quadrant I or II, \(\sin \theta\) is positive. Therefore, \(\sin \theta = \sqrt{1 - \frac{1}{\sec^2 \theta}}\).

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions commonly used in mathematics. It is the reciprocal of the cosine function. This means that

• if you know the cosine of an angle, you can easily find the secant by taking its reciprocal.
• Conversely, if you know the secant, you can find the cosine by taking its reciprocal as well.

For any angle \( \theta \), the secant is defined as \( \sec \theta = \frac{1}{\cos \theta} \).
Understanding the secant function is important because it allows us to solve many trigonometric problems. In particular, it can be useful when manipulating and transforming trigonometric expressions.
In this specific exercise, knowing that \( \sec \theta = \frac{1}{\cos \theta} \) helps us express \( \cos \theta \) in terms of \( \sec \theta \) and eventually find \( \sin \theta \). This relationship is important because it shows how different trigonometric functions interconnect, making it easier to solve complex problems.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone identities in trigonometry. It states that for any angle \( \theta \), the relationship \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is derived from the Pythagorean theorem and is valid for all angles, not just right triangles but also for any angle in a unit circle.
This identity helps us express one trigonometric function in terms of another. For example, if you know the value of \( \cos^2 \theta \), you can simply rearrange the identity to find \( \sin^2 \theta \) as \( \sin^2 \theta = 1 - \cos^2 \theta \).
In this exercise, replacing \( \cos^2 \theta = \frac{1}{\sec^2 \theta} \) in the Pythagorean identity allows us to discover \( \sin^2 \theta \) in terms of \( \sec \theta \), which is crucial to finding \( \sin \theta \) as required. This illustrates how trigonometric identities simplify complex expressions, making them easier to handle.

Quadrants in Trigonometry

Understanding the quadrants in trigonometry is vital for determining the sign of trigonometric functions. The coordinate plane is divided into four quadrants:

• In Quadrant I, both sine and cosine are positive.
• In Quadrant II, sine is positive, while cosine is negative.
• In Quadrant III, both sine and cosine are negative.
• In Quadrant IV, sine is negative, and cosine is positive.

Knowing in which quadrant an angle lies helps determine the sign of trigonometric functions like \( \sin \theta \) or \( \cos \theta \).
In the given exercise, since \( \theta \) lies in either Quadrant I or II, \( \sin \theta \) should be positive. This is crucial when taking the square root to find \( \sin \theta \) because the square root of a square results in both positive and negative solutions. By knowing the quadrant, we determine that \( \sin \theta = \sqrt{1 - \frac{1}{\sec^2 \theta}} \), as it must be the positive value based on its position in the quadrants. This understanding is essential for accurately solving trigonometric equations and ensuring your answer fits the given conditions.