Problem 44 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\csc \theta = -\sqrt{1 + \cot^2 \theta}\) in Quadrant III.

Step 1: Understand the Relationships between Trigonometric Functions

The cosecant function, \(\csc \theta\), is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\). The cotangent function, \(\cot \theta\), is the reciprocal of the tangent function, so \(\cot \theta = \frac{1}{\tan \theta}\). Moreover, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Therefore, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Understanding these relationships is essential for rewriting \(\csc \theta\) in terms of \(\cot \theta\).

Step 2: Express \(\sin \theta\) in terms of \(\cot \theta\)

Start with the definition of \(\cot \theta\): \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). In Quadrant III, both sine and cosine are negative, so \(\cos \theta = \cot \theta \cdot \sin \theta\). Rearrange this equation: \(\sin \theta = \frac{\cos \theta}{\cot \theta}\).

Step 3: Relate \(\cot \theta\) to a Known Identity

Using the Pythagorean identity \( \cos^2 \theta = 1 - \sin^2 \theta\), substitute \(\sin \theta\) from Step 2: \(\cos^2 \theta = 1 - \left(\frac{\cos \theta}{\cot \theta}\right)^2\).\

Step 4: Express \(\csc \theta\) in Terms of \(\cot \theta\)

Use the expression for \(\sin \theta\) to find \(\csc \theta\): \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\cos \theta}{\cot \theta}} = \frac{\cot \theta}{\cos \theta}\). In Quadrant III, simplify: \(\csc \theta = -\sqrt{1 + \cot^2 \theta}\).

Step 5: Conclude the Simplification

Since \(\cos \theta\) is negative in Quadrant III, the simple form for \(\csc \theta = -\sqrt{1 + \cot^2 \theta}\) aligns with the sign adjustments inherent to the quadrant.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This means that the cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the opposite side. In terms of the unit circle, it represents the inverse of the vertical distance from the origin to the angle's terminal side.
It’s important to note that because it depends on the sine function, it inherits its domain limitations—cosecant is undefined wherever sine is zero.
When dealing with angles in different quadrants, the sign of \( \csc \theta \) changes based on the sine's sign. For example, in Quadrant III, where \( \theta \) is located in this particular problem, sine is negative, making cosecant negative as well.

Cotangent Function

The cotangent function is another trigonometric function which is complementary but not as commonly used as sine or cosine. It is defined as the reciprocal of the tangent function:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Cotangent represents the ratio of the adjacent side to the opposite side in a right triangle.
This function can also be expressed using the sine and cosine functions, making it particularly useful in converting complex trigonometric expressions into simpler forms.
In Quadrant III, both sine and cosine are negative, making tangent and cotangent positive, as evident from their definitions. Understanding these relationships is key to manipulating trigonometric expressions and solving problems efficiently.

Quadrant III

When we refer to Quadrant III in the context of the unit circle, we are talking about angles between 180° and 270° (or \(\pi\) to \(\frac{3\pi}{2}\) radians). This section is crucial because it affects the signs of trigonometric functions.

• In Quadrant III, both the sine and cosine functions are negative.
• As a result, tangent and cotangent, which are ratios of sine and cosine, are positive.

Recognizing which quadrant an angle falls into allows you to correctly apply sign changes to your calculations. For instance, if you know \(\theta\) is in Quadrant III, you immediately adjust the signs of your trigonometric calculations accordingly.
This quadrant-specific sign information is essential for correctly evaluating expressions like \(\csc \theta\) and \(\cot \theta\) when solving trigonometric identities or simplifying expressions.