Problem 24 - Precalculus Essentials

Question

In Exercises \(23-34\), find the exact value of each of the remaining trigonometric functions of \(\theta .\) $$\sin \theta=-H, \quad \theta \text { in quadrant III }$$

Step-by-Step Solution

Answer

The exact values of the remaining trigonometric functions are: \(\cos \theta = -\sqrt{1 - H^2}\), \(\tan \theta = \frac{H}{\sqrt{1 - H^2}}\), \(\csc \theta = -\frac{1}{H}\), \(\sec \theta = -\frac{1}{\sqrt{1 - H^2}}\), and \(\cot \theta = \frac{\sqrt{1 - H^2}}{H}\)

Step 1: Find the value of \(\cos \theta\)

To find the value of \(\cos \theta\) we rearrange the Pythagorean identity as follows: \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). However, since we are in quadrant III where \(\cos \theta\) is negative, we get \(\cos \theta = -\sqrt{1 - H^2}\). This result comes from the fact that we are given \(\sin \theta = -H\).

Step 2: Find the value of \(\tan \theta\)

The tangent function is defined as the ratio of sine to cosine, so we have \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the known values we get \(\tan \theta = \frac{-H}{-\sqrt{1 - H^2}} = \frac{H}{\sqrt{1 - H^2}}\).

Step 3: Find the values of the reciprocal functions

The reciprocal of sine is cosecant (\(\csc \theta\)), the reciprocal of cosine is secant (\(\sec \theta\)), and the reciprocal of tangent is cotangent (\(\cot \theta\)). Therefore, \(\csc \theta = -\frac{1}{H}\), \(\sec \theta = -\frac{1}{\sqrt{1 - H^2}}\) and \(\cot \theta = \frac{\sqrt{1 - H^2}}{H}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It relates the square of the sine function and the square of the cosine function to 1: \[\sin^2(\theta) + \cos^2(\theta) = 1.\]

This equation stems from the Pythagorean theorem applied to a right-angled triangle, where the hypotenuse is length 1. If we have the value of the sine of an angle, like in the exercise where \(\sin \theta = -H\), we can rearrange the Pythagorean identity to find the cosine of that angle. Since \(\sin \theta\) is negative in the third quadrant, the corresponding \(\cos \theta\) must also be negative, resulting in \(\cos \theta\) being \(-\sqrt{1 - H^2}\).

Using Pythagorean Identity with Negative Values

Particularly in quadrant III, where both sine and cosine are negative, this identity helps us keep the signs straight for these trigonometric functions. Applying the identity correctly is crucial because it affects the computation of the other trigonometric functions, preserving the integrity of the solutions.

Tangent Function

The tangent function is another trigonometric function defined as the ratio of the sine of an angle to the cosine of the angle: \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.\]

In the exercise, after determining that \(\cos \theta = -\sqrt{1 - H^2}\), we apply this definition to find \(\tan \theta\) as \(\frac{-H}{-\sqrt{1 - H^2}}\), which simplifies to \(\frac{H}{\sqrt{1 - H^2}}\). It's important to note that in the third quadrant, where both sine and cosine are negative, the tangent, being the ratio of these two, will be positive.

Characteristics of the Tangent Function

The tangent function has some unique properties - it is periodic with period \(\pi\), unlike sine and cosine which have periods of \(2\pi\), and it has asymptotes where the cosine function is zero. This periodic and undefined nature at certain angles is essential to understand when studying the behavior of the tangent function.

Reciprocal Trigonometric Functions

In trigonometry, we also work with reciprocal functions, which are simply the inverses of the sine, cosine, and tangent functions. The reciprocal of the sine function is the cosecant (\(\csc\)), cosine's reciprocal is the secant (\(\sec\)), and tangent's reciprocal is the cotangent (\(\cot\)).

The formulas for these are:\[\csc(\theta) = \frac{1}{\sin(\theta)}, \ \sec(\theta) = \frac{1}{\cos(\theta)}, \ \cot(\theta) = \frac{1}{\tan(\theta)}.\]
For the given problem, after establishing the values for sine and cosine, the reciprocals can be found. The negative sign of sine and cosine in the third quadrant means that their reciprocals, cosecant and secant, will also be negative. This leads to \(\csc \theta = -\frac{1}{H}\) and \(\sec \theta = -\frac{1}{\sqrt{1 - H^2}}\), while \(\cot \theta\) is the reciprocal of the tangent, resulting in \(\cot \theta = \frac{\sqrt{1 - H^2}}{H}\).

Understanding Reciprocals in Various Quadrants

It's important to keep in mind the quadrant in which the angle resides because this determines the sign of the reciprocal functions. This concept helps extend understanding beyond the primary functions to their inverses, enriching a student's capability to solve complex trigonometric problems.