Problem 27 - Precalculus Essentials

Question

In Exercises \(23-34\), find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\cos \theta=\hat{H}, \quad 270^{\circ}<\theta<360^{\circ}$$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions of \( \theta \) are: \( \cos \theta = \sqrt{H} \), \( \sin \theta = -\sqrt{1 - H} \), \( \tan \theta = -\sqrt{\frac{1 - H}{H}} \), \( \sec \theta = \frac{1}{\sqrt{H}} \), \( \csc \theta = -\frac{1}{\sqrt{1 - H}} \), \( \cot \theta = -\sqrt{\frac{H}{1 - H}} \).

Step 1: Find \( \sin \theta \)

We're given \( \cos \theta =\sqrt{H} \). From the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can derive \( \sin \theta \). Substitute the given value of \( \cos \theta \): \( \sin^2 \theta = 1 - H \). Now, because \( 270^{\circ} < \theta < 360^{\circ} \), \( \sin \theta \) is negative in the 4th quadrant. So, \( \sin \theta = -\sqrt{1 - H} \).

Step 2: Find \( \tan \theta \)

Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), substitute the values obtained in Step 1 to get: \( \tan \theta = \frac{-\sqrt{1 - H}}{\sqrt{H}} = -\sqrt{\frac{1 - H}{H}} \).

Step 3: Find the Reciprocal Functions

Use the reciprocal identities to find the other three functions: \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\sqrt{H}} \), \( \csc \theta = \frac{1}{\sin \theta} = -\frac{1}{\sqrt{1 - H}} \), and \( \cot \theta = \frac{1}{\tan \theta} = -\sqrt{\frac{H}{1 - H}} \).

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental relationships in trigonometry. It expresses a connection between the squares of sine and cosine. The identity is given by the equation: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This identity essentially means that if you know one of these squares, you can easily find the other. For instance, if you know \( \cos \theta \), you can find \( \sin \theta \) using: \[ \sin^2 \theta = 1 - \cos^2 \theta \] This relationship is crucial when solving problems involving right-angled triangles or angles on the unit circle. In such contexts, it simplifies finding unknown trigonometric values given partial information.

trigonometric identities

Trigonometric identities are formulas or equations that hold true for all values of the variables where the expressions are defined. They're like secret codes in the world of mathematics that let you solve or transform complex trigonometric expressions into something simpler. Some of the most common trigonometric identities include:

• Reciprocal Identities
• Quotient Identities
• Co-Function Identities

Each type uses the basic trigonometric functions: sine, cosine, and tangent, and their reciprocal counterparts: cosecant, secant, and cotangent. These identities are essential tools not only in trigonometry class but also in calculus and engineering fields.

fourth quadrant

The fourth quadrant is one of the four sections of the coordinate plane. It is the lower right part, defined by angles between 270° and 360°, or equivalently between \( -90^\circ \) and 0° in radians. When an angle, like in our exercise, falls into this quadrant, special rules apply:

• \( \cos \theta \) is positive
• \( \sin \theta \) is negative
• \( \tan \theta \) is negative

Understanding where an angle resides in the coordinate plane can give insights into the sign and value of the trigonometric functions for these angles, thereby ensuring accurate solutions.

reciprocal functions

Reciprocal functions are trigonometric functions that are essentially the flips of sine, cosine, and tangent. These include cosecant, secant, and cotangent:

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

These functions are useful for solving various types of problems where inverses of standard trigonometric functions are required. They're most frequently used in trigonometric proofs and identities to provide alternate expressions for complex equations, simplifying the mathematical process.