Problem 26 - Precalculus Essentials

Question

In Exercises \(23-34\), find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\cos \theta=\frac{4}{5}, \quad \theta \text { in quadrant IV }$$

Step-by-Step Solution

Answer

\(\sin \theta = - \frac{3}{5}, \tan \theta = - \frac{3}{4}, \csc \theta = - \frac{5}{3}, \sec \theta = \frac{5}{4}, \cot \theta = - \frac{4}{3}\)

Step 1: Identify the Known and Apply the Pythagorean identity

We are given that, \(\cos \theta = \frac{4}{5}\) and \(\theta\) is in the fourth quadrant. We know that in the fourth quadrant, cosine is positive and sine and tangent are negative. The Pythagorean identity is \(\sin^2 \theta +\cos^2 \theta =1\). Rewritten for sin, this gives \(\sin \theta = \pm \sqrt{1 - \cos^2 \theta}\). This is applied here as \(\sin \theta = \pm \sqrt{1 - \left(\frac{4}{5}\right)^2}\). Considering the sine is negative in the fourth quadrant, \(\sin \theta = - \frac{3}{5}\).

Step 2: Find remaining trigonometric values

Next, use the known sine and cosine values to find the exact values of the remaining trigonometric functions. \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3/5}{4/5} = - \frac{3}{4}\), \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-3/5} = - \frac{5}{3}\), \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{4/5} = \frac{5}{4}\), \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-3/4} = - \frac{4}{3}\).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is essential for working with trigonometric equations and finding trigonometric values. The identity is an extension of the famous Pythagorean theorem which relates the sides of a right triangle. In trigonometry, the Pythagorean identity is written as \(\sin^2 \theta + \cos^2 \theta = 1\). This relationship is a cornerstone in trigonometry because it connects two fundamental trigonometric functions: sine \(\sin\) and cosine \(\cos\). It helps us find one trigonometric function value knowing the other.

For example, if we know the cosine of an angle \(\theta\), as \(\cos \theta = \frac{4}{5}\), we can find the sine by rearranging the identity as \(\sin \theta = \pm \sqrt{1 - \cos^2 \theta} = \pm \sqrt{1 - \left(\frac{4}{5}\right)^2} = \pm \sqrt{1 - \frac{16}{25}} = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5}\). In this scenario, sine would take a negative value in the fourth quadrant, indicating that the angle lies in a position where the y-coordinates of the unit circle are negative.

Trigonometric Values

Trigonometric values are the numerical values that represent the ratios of the sides of a right triangle or points on the unit circle corresponding to specific angles. Sine \(\sin\), cosine \(\cos\), tangent \(\tan\), cosecant \(\csc\), secant \(\sec\), and cotangent \(\cot\) are the six basic trigonometric functions.

Each function has a specific meaning and calculation:

• Sine represents the ratio of the opposite side to the hypotenuse.
• Cosine represents the ratio of the adjacent side to the hypotenuse.
• Tangent is the ratio of the sine to the cosine values of the angle.
• Cosecant, secant, and cotangent are respectively the reciprocals of sine, cosine, and tangent.

Using the given cos value, other function values are determined as: \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3/5}{4/5} = - \frac{3}{4}\), \(\csc \theta = \frac{1}{\sin \theta} = -\frac{5}{3}\), \(\sec \theta = \frac{1}{\cos \theta} = \frac{5}{4}\), and \(\cot \theta = \frac{1}{\tan \theta} = -\frac{4}{3}\). These trigonometric values are used to analyze various properties of triangles and circles, solve equations, and model periodic phenomena.

Quadrant Identification

Quadrant identification is the process of determining which of the four quadrants a given angle \(\theta\) resides in on the Cartesian coordinate system. This is important because it affects the signs (positive or negative) of trigonometric functions for that angle. Here's a brief rundown:

• Quadrant I: All trigonometric functions are positive here.
• Quadrant II: Sine \(\sin\) and cosecant \(\csc\) are positive, others are negative.
• Quadrant III: Tangent \(\tan\) and cotangent \(\cot\) are positive, others are negative.
• Quadrant IV: Cosine \(\cos\) and secant \(\sec\) are positive, while sine \(\sin\), tangent \(\tan\), cosecant \(\csc\), and cotangent \(\cot\) are negative.

In the given exercise, since \(\theta\) is in quadrant IV and \(\cos \theta\) is positive, we conclude that \(\sin \theta\) and \(\tan \theta\) must be negative as per the rule for the fourth quadrant. Thus, knowing the quadrant in which an angle lies allows us to correctly assign signs to the trigonometric values derived from the functions.