Problem 40 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\( \cot \theta = -\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \) in Quadrant II.

Step 1: Understand the Relationship Between Functions

The cotangent of an angle, \( \cot \theta \), is the reciprocal of its tangent: \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can express \( \cot \theta \) as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, we need to find \( \cos \theta \) in terms of \( \sin \theta \).

Step 2: Use the Pythagorean Identity

The Pythagorean identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \). Rearrange to solve for \( \cos \theta \): \( \cos^2 \theta = 1 - \sin^2 \theta \). Take the square root to find \( \cos \theta \): \( \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \).

Step 3: Determine the Sign of Cosine in Quadrant II

In Quadrant II, sine is positive and cosine is negative. Thus, \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \).

Step 4: Substitute to Express \( \cot \theta \) in Terms of \( \sin \theta \)

Substitute \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \) into the expression for \( \cot \theta \): \( \cot \theta = \frac{-\sqrt{1 - \sin^2 \theta}}{\sin \theta} \). This gives us \( \cot \theta = -\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \).

Key Concepts

Cotangent

Cotangent is one of the six main trigonometric functions and is closely related to tangent. If you know the tangent function, then understanding cotangent becomes a bit easier. The formula for cotangent is given by \( \cot \theta = \frac{1}{\tan \theta} \). Since tangent is defined as the ratio of sine to cosine, or \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), cotangent can also be expressed as the ratio of cosine to sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
This means that the value of cotangent is dependent on both sine and cosine values. Cotangent is undefined when sine is zero because you cannot divide by zero. This occurs when the angle \( \theta \) is a multiple of \( \pi \) (or 180 degrees). Understanding cotangent is essential for solving various trigonometric problems, especially those dealing with inverse functions and angles located in different quadrants.
Cotangent is useful when dealing with triangles and helps in identifying slopes and inclines in real-world applications such as engineering and physics.

Quadrant II

In trigonometry, the coordinate plane is divided into four quadrants. Quadrant II is the region where angles range from 90 degrees to 180 degrees, or \( \frac{\pi}{2} \) to \( \pi \) radians. In this quadrant, the sine function is positive, while the cosine and tangent functions are negative.
Knowing the sign of trigonometric functions in each quadrant is crucial for correctly evaluating and simplifying expressions. In Quadrant II:

• Sine (\( \sin \theta \)) is positive because the y-coordinate (opposite side of the angle) is positive in this part of the Cartesian plane.
• Cosine (\( \cos \theta \)) is negative, as the x-coordinate (adjacent side of the angle) is negative.
• Tangent (\( \tan \theta \)) is negative because it is the ratio of sine to cosine, i.e., \( \frac{\text{positive}}{\text{negative}} \).

The characteristics of these functions in Quadrant II help in determining the behavior of derived functions like cotangent, as seen in trigonometric identities and equations.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, represented as \( \sin^2 \theta + \cos^2 \theta = 1 \). It states that the square of the sine of an angle, added to the square of the cosine of the same angle, equals 1. This identity is critical because it allows us to express one trigonometric function in terms of another.
For example, you can solve for \( \cos \theta \) if you know \( \sin \theta \), using the Pythagorean identity: \( \cos^2 \theta = 1 - \sin^2 \theta \). By taking the square root, we find \( \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \).
The sign of \( \cos \theta \) will depend on the quadrant where the angle \( \theta \) is located. In Quadrant II, \( \cos \theta \) is negative, simplifying to \( \cos \theta = -\sqrt{1 - \sin^2 \theta} \). This step is crucial in converting trigonometric expressions as it bridges between functions.

Sin Theta

Sine, denoted as \( \sin \theta \), is a primary trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle. It is one of the key functions in understanding unit circle properties and solving trigonometric equations.
In the context of the given problem, sine plays a crucial role because it helps determine other related functions in different quadrants. Knowing \( \sin \theta \) is especially useful in expressing other trigonometric functions like cosine and cotangent. When \( \theta \) is in Quadrant II, \( \sin \theta \) remains positive, which helps in assessing the implications on cosine and other associated functions.
Sine function is pivotal in a multitude of calculations in physics, engineering, and geometry, among other fields, due to its role in wave equations and circular motion. It's essential to understand \( \sin \theta \) for tackling any comprehensive trigonometric problem.