Problem 128 - A Graphical Approach to Precalculus with Limits

Question

Use fundamental identities to find each expression. Write \(\tan \theta\) in terms of \(\cos \theta\) if \(\theta\) is in quadrant IV.

Step-by-Step Solution

Answer

\( \tan \theta = \frac{-\sqrt{1 - \cos^2 \theta}}{\cos \theta} \) in Quadrant IV.

Step 1: Identify Trigonometric Identities

We know from the basic trigonometric identities that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). In order to express \( \tan \theta \) in terms of \( \cos \theta \), we need to find \( \sin \theta \).

Step 2: Use Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Since we need to express everything in terms of \( \cos \theta \), we rearrange it to find \( \sin \theta \): \( \sin \theta = \pm \sqrt{1 - \cos^2 \theta} \).

Step 3: Determine the Sign of \( \sin \theta \)

Since \( \theta \) is in Quadrant IV, we know that \( \sin \theta \) is negative. Therefore, \( \sin \theta = -\sqrt{1 - \cos^2 \theta} \) in this quadrant.

Step 4: Substitute \( \sin \theta \) into the Tangent Formula

Substitute \( \sin \theta = -\sqrt{1 - \cos^2 \theta} \) into the tangent identity: \( \tan \theta = \frac{-\sqrt{1 - \cos^2 \theta}}{\cos \theta} \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It relates the angles and sides of a right triangle, but it's also defined using the sine and cosine functions. In the context of trigonometric identities, the relationship between these functions is given by:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This identity shows how \( \tan \theta \) is the ratio of the opposite side to the adjacent side in a right triangle, or sine over cosine. For many problems, especially those involving different quadrants, expressing \( \tan \theta \) using other function values like \( \cos \theta \) becomes essential. Breaking down the tangent function into these components allows for more versatile manipulation and understanding of trigonometric equations.
Ultimately, the tangent function helps bridge various trigonometric scenarios, whether it is solving equations or analyzing periodic behaviors.

Pythagorean Identity

The Pythagorean Identity is a crucial element in understanding trigonometric functions, often serving as a stepping stone in derivations and proofs. The Pythagorean Identity is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the Pythagorean Theorem, representing the squared relationships among the triangle sides. By knowing either \( \sin \theta \) or \( \cos \theta \), you can determine the other. For example:

• If you have a known \( \cos \theta \), you can rearrange the identity to find \( \sin \theta \) as \( \sin \theta = \pm \sqrt{1 - \cos^2 \theta} \).

This rearrangement becomes particularly useful when needing to express \( \tan \theta \) in terms of \( \cos \theta \). It's essential to understand that the sign of \( \sin \theta \) depends on the quadrant in which the angle resides. The Pythagorean Identity not only confirms the intrinsic relationships of sine and cosine but also allows the expansion of expressions needed for solving trigonometric problems.

Quadrant IV

Understanding the quadrant in which an angle resides is crucial for determining the signs of trigonometric functions. In Quadrant IV, specific behaviors of functions must be considered:

• \( \sin \theta \) is negative because the sine function represents the y-coordinate, which is below the x-axis in this quadrant.
• \( \cos \theta \) is positive since it represents the x-coordinate, aligning with the positive x-axis.
• \( \tan \theta \), being the ratio \( \frac{\sin \theta}{\cos \theta} \), is negative because a negative value (\( \sin \theta \)) over a positive value (\( \cos \theta \)) results in a negative outcome.

The determination of signs in Quadrant IV is vital because it influences the final expression of any trigonometric function derived in terms of another. Specifically, when expressing \( \tan \theta \) using identities and known values, recognizing the quadrant ensures accurate results. In this quadrant, the negative sign assists in aligning the derived expressions with the real geometric representation of angle \( \theta \).