Problem 52 - Precalculus Mathematics for Calculus

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\tan \theta=-4, \quad \sin \theta>0$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{4}{\sqrt{17}}, \cos \theta = -\frac{1}{\sqrt{17}}, \tan \theta = -4, \csc \theta = \frac{\sqrt{17}}{4}, \sec \theta = -\sqrt{17}, \cot \theta = -\frac{1}{4} \).

Step 1: Understanding the Given Information

We are given that \( \tan \theta = -4 \) and \( \sin \theta > 0 \). The tangent is negative, which means \( \theta \) is in either the second or fourth quadrant. However, since \( \sin \theta > 0 \), \( \theta \) must be in the second quadrant.

Step 2: Using Pythagorean Identity for \\tan

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Since \( \tan \theta = -4 \), this implies \( \frac{\sin \theta}{\cos \theta} = -4 \). In the second quadrant, \( \sin \theta \) is positive, and \( \cos \theta \) is negative.

Step 3: Setting Up an Expression for \\sin \\theta and \\cos \\theta

Let \( \sin \theta = 4k \) and \( \cos \theta = -k \). The ratio becomes \( \frac{4k}{-k} = -4 \), which matches the given \( \tan \theta \).

Step 4: Applying Pythagorean Identity

Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), substitute the expressions for \( \sin \theta \) and \( \cos \theta \): \( (4k)^2 + (-k)^2 = 1 \).

Step 5: Simplifying the Equation

Simplify \( 16k^2 + k^2 = 1 \) to get \( 17k^2 = 1 \). Thus, \( k^2 = \frac{1}{17} \) and \( k = \frac{1}{\sqrt{17}} \).

Step 6: Finding \\sin \\theta and \\cos \\theta

Since \( \sin \theta = 4k \), \( \sin \theta = \frac{4}{\sqrt{17}} \). For \( \cos \theta = -k \), \( \cos \theta = -\frac{1}{\sqrt{17}} \).

Step 7: Calculating Remaining Trigonometric Functions

Now, calculate \( \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{17}}{4} \), \( \sec \theta = \frac{1}{\cos \theta} = -\sqrt{17} \), and \( \cot \theta = \frac{1}{\tan \theta} = -\frac{1}{4} \).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental relationship in trigonometry, connecting the squares of the sine and cosine functions. It states that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity derives from the Pythagorean theorem applied to a unit circle, where the radius is 1. In practical usage, knowing this identity allows you to find trigonometric function values if you have either sine or cosine. For example, knowing \( \sin \theta \), you can rearrange the identity to determine \( \cos \theta \):

• \( \cos^2 \theta = 1 - \sin^2 \theta \)
• \( \cos \theta = \pm \sqrt{1 - \sin^2 \theta} \)

It helps particularly when calculating or solving trigonometric problems in exercises like finding the value of \( \theta \). By setting \( \sin \theta \) and \( \cos \theta \) to expressed variables, as in the given exercise, you directly utilize this identity to solve for unknowns and confirm each function value's validity.

quadrants of the unit circle

Understanding the quadrants of the unit circle is crucial for determining the sign and value of trigonometric functions. A unit circle is divided into four quadrants, each contributing to different signs for the sine, cosine, and tangent functions.

• First Quadrant: \( \sin \theta > 0 \), \( \cos \theta > 0 \), \( \tan \theta > 0 \)
• Second Quadrant: \( \sin \theta > 0 \), \( \cos \theta < 0 \), \( \tan \theta < 0 \)
• Third Quadrant: \( \sin \theta < 0 \), \( \cos \theta < 0 \), \( \tan \theta > 0 \)
• Fourth Quadrant: \( \sin \theta < 0 \), \( \cos \theta > 0 \), \( \tan \theta < 0 \)

Each quadrant reflects the sign combination of different trigonometric functions. In the prescribed exercise, knowing \( \tan \theta = -4 \) (negative) and \( \sin \theta > 0 \), guides us to conclude that \( \theta \) lies in the second quadrant. This knowledge assists in setting appropriate constraints on the function values when using the Pythagorean identity and related strategies.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable angle. They serve as essential tools for transforming and simplifying trigonometric expressions in mathematics. Some common trigonometric identities include:

• Pythagorean Identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal Identities: e.g., \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \)
• Tangent and Cotangent Identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In the given problem, the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is initially utilized to express \( \tan \theta \) in terms of sine and cosine. Using the Pythagorean identity further allows calculation of exact trigonometric values. Together, these identities guide the step-by-step solution process to find specific trigonometric function values of \( \theta \), illuminating their interconnected nature.