Problem 124 - A Graphical Approach to Precalculus with Limits

Question

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). $$\csc \theta=-3, \text { given that } \cos \theta>0$$

Step-by-Step Solution

Answer

\( \sin \theta = -\frac{1}{3} \), \( \cos \theta = \frac{2\sqrt{2}}{3} \), \( \tan \theta = -\frac{\sqrt{2}}{4} \), \( \csc \theta = -3 \), \( \sec \theta = \frac{3\sqrt{2}}{4} \), \( \cot \theta = -2\sqrt{2} \).

Step 1: Understanding Cosecant and Its Implications

The cosecant function, \( \csc \theta \), is the reciprocal of the sine function. Therefore, \( \csc \theta = -3 \) means that \( \sin \theta = -\frac{1}{3} \). Since sine is negative, \( \theta \) must be in either the third or fourth quadrant. However, we are given that \( \cos \theta > 0 \), which restricts \( \theta \) to the fourth quadrant.

Step 2: Use Pythagorean Identity to Find Cosine

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), substitute \( \sin \theta = -\frac{1}{3} \):\[(-\frac{1}{3})^2 + \cos^2 \theta = 1\]\[\frac{1}{9} + \cos^2 \theta = 1\]Subtract \( \frac{1}{9} \) from 1:\[\cos^2 \theta = \frac{8}{9}\]Since \( \cos \theta > 0 \), take the positive square root:\[\cos \theta = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}\]

Step 3: Find Tangent Using Sine and Cosine

The tangent function is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the known values:\[\tan \theta = \frac{-\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{4}\]

Step 4: Determine Remaining Trigonometric Functions

The remaining trigonometric functions are secant and cotangent. For secant, which is the reciprocal of cosine:\[\sec \theta = \frac{1}{\cos \theta} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}\]For cotangent, which is the reciprocal of tangent:\[\cot \theta = \frac{1}{\tan \theta} = -2\sqrt{2}\]

Step 5: Summary of All Trigonometric Values

Compile all the trigonometric values:- \( \sin \theta = -\frac{1}{3} \)- \( \cos \theta = \frac{2\sqrt{2}}{3} \)- \( \tan \theta = -\frac{\sqrt{2}}{4} \)- \( \csc \theta = -3 \)- \( \sec \theta = \frac{3\sqrt{2}}{4} \)- \( \cot \theta = -2\sqrt{2} \)

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is a less commonly used trigonometric function, but it can be quite useful. Essentially, it is the reciprocal of the sine function. This means that for any angle \( \theta \), \( \csc \theta = \frac{1}{\sin \theta} \).
So, when we say \( \csc \theta = -3 \), we are implying that \( \sin \theta = -\frac{1}{3} \). Understanding this reciprocal relationship can help simplify trigonometric problems.

• If \( \csc \theta \) is negative, like \( -3 \), then \( \sin \theta \) is negative.
• In the unit circle, sine is negative in the third and fourth quadrants. However, given \( \cos \theta > 0 \), \( \theta \) must be specifically in the fourth quadrant.

Recognizing these quadrant rules is crucial for applying trigonometric identities correctly and verifies the possible values for \( \theta \).

Pythagorean Identity

The Pythagorean identity is foundational in trigonometry. It states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity arises from the Pythagorean theorem applied to a unit circle, an insightful way to link algebraic and geometric interpretations.
When given \( \sin \theta = -\frac{1}{3} \), use this identity to find \( \cos \theta \):
- Substitute into the identity: \( \sin^2 \theta = \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \).
- Therefore, \( \frac{1}{9} + \cos^2 \theta = 1 \). Subtract \( \frac{1}{9} \) from 1 to isolate \( \cos^2 \theta \): \( \cos^2 \theta = \frac{8}{9} \).
- Since \( \cos \theta > 0 \), take the positive square root resulting in \( \cos \theta = \frac{2\sqrt{2}}{3} \).
Understanding the application of this identity is essential for solving trigonometric functions when partial information is given, and it rhythmically connects the sine and cosine functions.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include cosecant, secant, and cotangent functions. Each corresponds to the reciprocal of the basic trigonometric ratios.
Understanding them enhances problem-solving strategies:

• The cosecant function is \( \csc \theta = \frac{1}{\sin \theta} \).
• The secant function is \( \sec \theta = \frac{1}{\cos \theta} \). For this example, if \( \cos \theta = \frac{2\sqrt{2}}{3} \), then \( \sec \theta = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4} \).
• The cotangent function is \( \cot \theta = \frac{1}{\tan \theta} \). If \( \tan \theta = -\frac{\sqrt{2}}{4} \), then \( \cot \theta = -2\sqrt{2} \).

Understanding these reciprocal relations allows you to derive other trigonometric values when just one is known, therefore, making it simpler to solve trigonometric equations efficiently.