Problem 44 - Algebra 2

Question

Find the value of each expression. $$ \csc \theta, \text { if } \cot \theta=-\frac{7}{12} ; 90^{\circ}<\theta<180^{\circ} $$

Step-by-Step Solution

Answer

\( \csc \theta = \frac{\sqrt{193}}{12} \)

Step 1: Understand the Problem

We need to find the value of \( \csc \theta \) given that \( \cot \theta = -\frac{7}{12} \) and \( 90^{\circ} < \theta < 180^{\circ} \). The angle \( \theta \) lies in the second quadrant.

Step 2: Use Trigonometric Identities

Recall the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, \( \cos \theta = -7k \) and \( \sin \theta = 12k \) for some constant \( k \).

Step 3: Apply Pythagorean Identity

Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute \( \sin \theta \) and \( \cos \theta \): \[ (12k)^2 + (-7k)^2 = 1. \] Simplifying gives \( 144k^2 + 49k^2 = 1 \), leading to \( 193k^2 = 1 \) and \( k^2 = \frac{1}{193} \).

Step 4: Calculate \( \sin \theta \) and \( \csc \theta \)

From \( k^2 = \frac{1}{193} \), we have \( k = \pm \frac{1}{\sqrt{193}} \). Since \( \theta \) is in the second quadrant, \( \sin \theta = 12k > 0 \) and \( k = \frac{1}{\sqrt{193}} \). Thus, \( \sin \theta = \frac{12}{\sqrt{193}} \), and \( \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{193}}{12} \).

Key Concepts

Cosecant in Trigonometry

Cosecant, abbreviated as \( \csc \theta \), is one of the basic trigonometric functions. It is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). Cosecant is less commonly used than sine, cosine, or tangent, but it is equally important.

• Reciprocal Relationship: Remember, cosecant and sine are inverses. If you know the sine of an angle, you can easily find the cosecant by simply taking its reciprocal.
• Undefined Values: It is important to note that \( \csc \theta \) is undefined for angles where the sine is zero, such as \( \theta = 0^{\circ}, 180^{\circ} \).

In problems like the one at hand, once we determine \( \sin \theta \), finding \( \csc \theta \) becomes straightforward by calculating \( \frac{1}{\sin \theta} \). This functional relationship is a cornerstone of solving trigonometric expressions.

Understanding Cotangent

Cotangent, denoted as \( \cot \theta \), is another important trigonometric function. It is the reciprocal of the tangent function, and in terms of sine and cosine, it is expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

• Relation to Tangent: Just like cosine is the reciprocal of sine, cotangent is simply \( \frac{1}{\tan \theta} \).
• Quadrant Considerations: The sign of \( \cot \theta \) depends on the quadrant in which \( \theta \) lies. This has implications for values, as functions might be negative or positive depending on this position.

In this problem, we are given \( \cot \theta = -\frac{7}{12} \), showing that the relationship between cosine and sine has a specific ratio that we harness to delve deeper into the trigonometric identity applications further on.

Pythagorean Identities

Pythagorean identities are fundamental tools in trigonometry that relate the squares of sine, cosine, and other trigonometric functions. The most basic form is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity helps find unknown trigonometric values when another is known.

• Expanding Relationships: Besides \( \sin^2 \theta + \cos^2 \theta = 1 \), there are other forms like \( 1 + \tan^2 \theta = \sec^2 \theta \) and \( 1 + \cot^2 \theta = \csc^2 \theta \).
• Substitution and Solving: This problem uses the identity to determine the values of \( \sin \theta \) and \( \cos \theta \) by substituting proportional values in terms of \( k \).

Understanding these identities facilitates solving trigonometric equations, allowing for substitution and manipulation of variables to unveil hidden relationships, just as seen in the given solution.

Trigonometry in the Second Quadrant

Trigonometric functions behave differently depending on which quadrant the angle \( \theta \) falls into on the unit circle.

• Functions Signs: In the second quadrant \((90^{\circ} < \theta < 180^{\circ})\), \( \sin \theta \) is positive while \( \cos \theta \) and \( \tan \theta \) are negative. This is because in this quadrant, the x-coordinates (which represent cosine) are negative, and the y-coordinates (which represent sine) are positive.
• Cosecant and Cotangent: Similarly, \( \csc \theta \) will be positive since it is the reciprocal of sine. \( \cot \theta \) will be negative as it relies on the sine and cosine proportion.

It's crucial to understand the quadrant-specific behavior since it influences the sign and magnitude of trigonometric ratios and their reciprocals. This knowledge is pivotal when evaluating these functions under specified angle constraints.