Problem 39 - Precalculus Mathematics for Calculus

Question

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$\tan \theta, \quad \cos \theta ; \quad \theta \text { in Quadrant III }$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Quadrant

Since the angle \(\theta\) is in the third quadrant, both sine and cosine values are negative, but tangent is positive. This is because tangent is the ratio of sine to cosine, \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), and a negative divided by a negative is positive.

Step 2: Recall the Basic Trigonometric Identity

The basic trigonometric identity we will use is \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). We will express \(\sin\theta\) in terms of \(\cos\theta\) using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\).

Step 3: Express \(\sin\theta\) in Terms of \(\cos\theta\)

Using the identity \(\sin^2\theta + \cos^2\theta = 1\), solve for \(\sin^2\theta\): \(\sin^2\theta = 1 - \cos^2\theta\). Thus, \(\sin\theta = -\sqrt{1 - \cos^2\theta}\) because \(\sin\theta\) is negative in the third quadrant.

Step 4: Express \(\tan\theta\) in Terms of \(\cos\theta\)

Substitute \(\sin\theta = -\sqrt{1 - \cos^2\theta}\) into the tangent formula: \(\tan\theta = \frac{-\sqrt{1 - \cos^2\theta}}{\cos\theta}\). Simplifying, we write \(\tan\theta = \frac{\sqrt{1 - \cos^2\theta}}{-\cos\theta}\), ensuring the sign is correct for Quadrant III.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, dealing with the ratio of two other essential trigonometric functions: sine and cosine. Specifically, it is defined as the ratio of the sine of an angle to the cosine of that same angle. In mathematical terms, this is expressed as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).

In any given quadrant of the unit circle, the signs of sine and cosine can change, which impacts the tangent.

• In Quadrant I, both sine and cosine are positive, making tangent positive.
• In Quadrant II, sine is positive while cosine is negative, making tangent negative.
• In Quadrant III, both sine and cosine are negative, thus making tangent positive again, as a negative divided by a negative yields a positive result.
• In Quadrant IV, sine is negative, and cosine is positive, resulting in a negative tangent.

Understanding this behavior in the various quadrants is crucial when solving trigonometric equations and understanding their properties in different contexts.

Cosine Function

The cosine function is another fundamental trigonometric function, essential in describing the relationship between the angles and sides of a right triangle. Cosine of an angle \(\theta\) is defined as the adjacent side over the hypotenuse when looking at a right triangle. Cosine is often abbreviated as \(\cos\).

On the unit circle, cosine represents the x-coordinate for a given angle \(\theta\). As the angle traverses through the quadrants:

• In Quadrant I, cosine values are positive.
• In Quadrant II, cosine turns negative, as the angles extend beyond 90 degrees, shifting the x-values to the left side of the unit circle.
• In Quadrant III, cosine remains negative, continuing this trend.
• In Quadrant IV, cosine becomes positive again, as angles circle back to positive x-values.

When analyzing trigonometric identities or problems, recognizing the sign and behavior of the cosine function in its respective quadrant helps in understanding and solving complex trigonometric identities and equations.

Trigonometric Identities

Trigonometric identities are essential equations involving trigonometric functions that hold true for all values of the included variables. They are crucial tools for simplifying and solving trigonometric expressions and equations.

One of the most fundamental identities involves the Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \). This identity is instrumental in expressing one trigonometric function in terms of another. For instance, in the context of the given exercise with \(\theta\) in Quadrant III, we solved for \(\sin^2\theta\) from this identity:

• \(\sin^2\theta = 1 - \cos^2\theta\)
• Then, \(\sin\theta = -\sqrt{1 - \cos^2\theta}\) since sine is negative in Quadrant III.

Additionally, identities such as \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) allow us to express the tangent function using known values of sine and cosine, demonstrating how interconnected these functions and identities are in trigonometry. These identities not only aid in solving problems but also deepen the understanding of the intrinsic relationships between trigonometric functions.