Problem 41 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

In Quadrant IV, \(\cos \theta = \sqrt{1 - \sin^2 \theta}\).

Step 1: Understand the characteristics of Quadrant IV

In Quadrant IV, the cosine function \(\cos \theta\) is positive, while the sine function \(\sin \theta\) is negative. We need to consider these signs when expressing \(\cos \theta\) in terms of \(\sin \theta\).

Step 2: Use the Pythagorean identity

Recall the Pythagorean identity for trigonometric functions: \(\sin^2 \theta + \cos^2 \theta = 1\). We can solve this identity for \(\cos \theta\) in terms of \(\sin \theta\).

Step 3: Isolate \(\cos \theta\)

Reorganize the Pythagorean identity to express \(\cos^2 \theta\): \(\cos^2 \theta = 1 - \sin^2 \theta\). Take the square root of both sides to solve for \(\cos \theta\).

Step 4: Consider the sign of \(\cos \theta\)

When taking the square root to find \(\cos \theta = \pm \sqrt{1 - \sin^2 \theta}\), we must consider the sign of \(\cos \theta\) in Quadrant IV. Since \(\cos \theta\) is positive in Quadrant IV, we choose \(\cos \theta = \sqrt{1 - \sin^2 \theta}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It relates the squares of the sine and cosine of an angle. The identity states: \( \sin^2 \theta + \cos^2 \theta = 1 \). This is similar to the Pythagorean theorem, hence the name. The identity always holds true for any angle \( \theta \) and is useful in various mathematical problems. It allows us to express one trigonometric function in terms of another. Step-by-step usage:

• Identify the known trigonometric function. In this case, either \( \sin \theta \) or \( \cos \theta \)
• Substitute the known value into the identity.
• Reorganize the equation to isolate the desired function.
• Take the square root if necessary, remembering to consider the sign depending on the quadrant.

By using this identity, you can always find \( \cos \theta \) if \( \sin \theta \) is known, and vice versa.

Trigonometric Functions Quadrant Signs

Trigonometric functions have specific signs depending on which quadrant the angle \( \theta \) is in. These signs are important when resolving trigonometric identities or calculations. Quadrants are divided into four sections around the origin of a coordinate plane. In each, certain trigonometric functions are positive or negative:

• Quadrant I: \( \sin \theta \), \( \cos \theta \), \( \tan \theta \) are positive.
• Quadrant II: \( \sin \theta \) positive; \( \cos \theta \), \( \tan \theta \) negative.
• Quadrant III: \( \tan \theta \) positive; \( \sin \theta \), \( \cos \theta \) negative.
• Quadrant IV: \( \cos \theta \) positive; \( \sin \theta \), \( \tan \theta \) negative.

For the given problem, knowing \( \theta \) is in Quadrant IV helps determine that \( \cos \theta \) is positive, and \( \sin \theta \) is negative, which is crucial when solving the identity.

Cosine and Sine Relationship

Cosine and sine are two primary trigonometric functions that describe different aspects of a right triangle or circle. The relationship between these two is not only defined by the Pythagorean identity, but their specific behaviors are also essential to understand in the context of trigonometry.

• Sine (\( \sin \theta \)) refers to the vertical component of a unit circle at angle \( \theta \).
• Cosine (\( \cos \theta \)) refers to the horizontal component of a unit circle at the same angle.

When these functions are considered within the unit circle, their values are between -1 and 1, explaining their trigonometric circle correlation. In the context of quadrants, as mentioned: - In Quadrant IV, \( \cos \theta \) remains positive, indicating it is the dominant component in this section of the circle. - Meanwhile, \( \sin \theta \) is negative, reflecting its downward movement away from the positive vertical axis. Understanding this helps validate calculated values of \( \cos \theta \) and \( \sin \theta \), ensuring their signs and magnitudes match their respective quadrant expectations.