Problem 29 - Precalculus: A Concise Course

Question

Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) with the given constraint. $$ \sec \theta=-2 \quad \sin \theta<0 $$

Step-by-Step Solution

Answer

The values of the six trigonometric functions are: \( \sin \theta = - \sqrt{3}/2 \), \( \cos \theta = -1/2 \), \( \tan \theta = \sqrt{3} \), \( \csc \theta = -2/\sqrt{3} \), \( \sec \theta = -2 \), \( \cot \theta = 1/\sqrt{3} \).

Step 1: Identify the Quadrant of \( \theta \)

Since \( \sec \theta = -2 \), this implies that \( \cos \theta = -1/2 \). A negative cos value suggests that \( \theta \) is either in the second or the third quadrant. But since \( \sin \theta < 0 \), \( \theta \) is in the third quadrant where both cosine and sine values are negative.

Step 2: Solve for \( \sin \theta \)

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to solve for \( \sin \theta \). Substituting \( \cos \theta = -1/2 \) into the equation and solving for \( \sin \theta \) yields \( \sin \theta = - \sqrt{3}/2 \).

Step 3: Solve for \( \tan \theta \)

Find \( \tan \theta \) using its definition in terms of sine and cosine: \( \tan \theta = \sin \theta / \cos \theta\). Substituting \(\sin \theta = - \sqrt{3}/2 \) and \( \cos \theta = -1/2 \), we find \( \tan \theta = \sqrt{3} \).

Step 4: Find the reciprocals

Following from above, we can find the remaining three trigonometric functions by taking the reciprocals: \( \csc \theta = -2/\sqrt{3} \), \( \sec \theta = -2 \), and \( \cot \theta = 1/\sqrt{3} \).

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is closely related to the cosine function and is defined as the reciprocal of the cosine. That is, \( \sec \theta = \frac{1}{\cos \theta} \). When dealing with the secant function, it's essential to know that it is undefined wherever the cosine is zero because division by zero is undefined.

• For example, if \( \sec \theta = -2 \), then \( \cos \theta = \frac{-1}{2} \).
• This indicates that \( \theta \) could potentially be in any quadrant where cosine is negative.

Because secant is the reciprocal of cosine, understanding the behavior of \( \cos \theta \) helps in determining the value of \( \sec \theta \) and can guide you to infer certain properties like identifying the quadrant in which the angle lies.

This is crucial in exercises dealing with constraints, as seen in the problem where the given \( \sec \theta = -2 \) hints towards \( \theta \) being either in the second or the third quadrant.

Pythagorean Identity

The Pythagorean identity is a foundational equation in trigonometry. It expresses the inherent relationship between sine and cosine:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity helps you relate different trigonometric functions when one is known. For example, when \( \cos \theta = \frac{-1}{2} \), you can plug this value into the identity to solve for the sine function:

• Substitute \( \cos \theta = \frac{-1}{2} \) into \( \sin^2 \theta + \left(\frac{-1}{2}\right)^2 = 1 \)
• This simplifies to \( \sin^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \)
• Solving gives \( \sin \theta = -\sqrt{\frac{3}{4}} = -\frac{\sqrt{3}}{2} \)

The Pythagorean identity is incredibly useful because it allows for the determination of missing values in trigonometric equations. This identity serves as the backbone of solving trigonometric problems where interrelations between sine and cosine are used to find unknown trigonometric values.

Trigonometric Quadrants

Understanding trigonometric quadrants is key to analyzing and solving trigonometric functions. Each quadrant on the Cartesian coordinate plane affects the signs of trigonometric functions.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Both sine and cosine are negative, but tangent is positive.
• Quadrant IV: Cosine is positive, while sine and tangent are negative.

In the original exercise, \( \theta \) is trapped between quadrants II and III due to the negative secant function, which points to a negative cosine value. However, the additional constraint \( \sin \theta < 0 \) narrows \( \theta \) down to the third quadrant.
In the context of the problem, interpreting the quadrant allows us to confirm which trigonometric signs should be used.This knowledge contributes significantly to correctly solving trigonometric exercises, especially in identifying the correct signs for calculated trigonometric values.