Problem 20 - Basic Technical Mathematics with Calculus

Question

Find the values of the indicated functions. In Exercises \(17-20,\) give answers in exact form. In Exercises \(21-24,\) the values are approximate. Given \(\sec \theta=\sqrt{5} / 2,\) find \(\tan \theta\) and \(\cos \theta\).

Step-by-Step Solution

Answer

\( \cos \theta = \frac{2\sqrt{5}}{5} \), \( \tan \theta = \pm \frac{1}{2} \).

Step 1: Understanding the Given Function

We are given that \( \sec \theta = \frac{\sqrt{5}}{2} \). The secant function is the reciprocal of the cosine function, so we can write:\[ \sec \theta = \frac{1}{\cos \theta} \] This means \( \cos \theta = \frac{2}{\sqrt{5}} \).

Step 2: Rationalizing the Cosine Value

To express \( \cos \theta = \frac{2}{\sqrt{5}} \) in exact form, we can rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{5} \). This gives us:\[ \cos \theta = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \].

Step 3: Defining the Trigonometric Identity for Tangent

We need to find \( \tan \theta \). We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). We can use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \).

Step 4: Finding Sine Using the Pythagorean Identity

Substitute the \( \cos \theta \) value into the Pythagorean identity:\[ \sin^2 \theta + \left(\frac{2\sqrt{5}}{5}\right)^2 = 1 \]Calculating the square puts us at \( \frac{20}{25} \), simplifying it gives \( \frac{4}{5} \). Thus, \( \sin^2 \theta = 1 - \frac{4}{5} = \frac{1}{5} \), meaning \( \sin \theta = \pm \frac{1}{\sqrt{5}} \). Rationalizing this gives, \( \sin \theta = \pm \frac{\sqrt{5}}{5} \).

Step 5: Calculating the Tangent of Theta

Using \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), substitute the values:\[ \tan \theta = \frac{\pm \frac{\sqrt{5}}{5}}{\frac{2\sqrt{5}}{5}} = \pm \frac{1}{2} \].

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function:

\[ \sec \theta = \frac{1}{\cos \theta} \]

This means that if you have the value of \( \sec \theta \), you can find \( \cos \theta \) by taking its reciprocal. For example, if \( \sec \theta = \frac{\sqrt{5}}{2} \), then \( \cos \theta = \frac{2}{\sqrt{5}} \). This relationship is crucial in solving trigonometric equations, as knowing one function can easily help determine another through such reciprocal identities.

In practice, if the given \( \sec \theta \) seems complex, it's often beneficial to work with rationalized forms to simplify calculations, especially when finding exact trigonometric values.

Cosine

The cosine function, symbolized as \( \cos \theta \), is a core component of trigonometry. It is defined on the unit circle as the horizontal coordinate of a point corresponding to an angle \( \theta \). In this context, having calculated \( \cos \theta = \frac{2}{\sqrt{5}} \), it is often preferable to express it in a more standard form by rationalizing the denominator. This involves multiplying the numerator and denominator by \( \sqrt{5} \) to obtain:

\[ \cos \theta = \frac{2\sqrt{5}}{5} \]

Cosine values can range between -1 and 1 on the unit circle, and they provide an insightful measure of how much the angle "leans" towards the x-axis.

Mastering cosine values and their transformations is fundamental when dealing with various identities and function transformations in trigonometry.

Tangent

The tangent function, noted as \( \tan \theta \), represents another important trigonometric ratio. It is the quotient of the sine and cosine functions:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

In scenarios where cosine and sine are known, calculating the tangent becomes straightforward. Using our example, with \( \sin \theta = \pm \frac{\sqrt{5}}{5} \) and \( \cos \theta = \frac{2\sqrt{5}}{5} \), the tangent can be found as:

\[ \tan \theta = \frac{\pm \frac{\sqrt{5}}{5}}{\frac{2\sqrt{5}}{5}} = \pm \frac{1}{2} \]

Tangent values describe the angle's slope when interpreted as a line on the Cartesian plane. It is particularly useful when assessing steepness or angle measures, especially in practical applications like physics or engineering.

Pythagorean Identity

One of the most powerful tools in trigonometry is the Pythagorean Identity, which relates the square of sin and cos functions:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

This identity helps us discover unknown trigonometric values when we know others. Substitute the known \( \cos \theta = \frac{2\sqrt{5}}{5} \) into the identity to find \( \sin \theta \):

First, square the cosine:
\[ \left(\frac{2\sqrt{5}}{5}\right)^2 = \frac{20}{25} = \frac{4}{5} \]

Then, solve for \( \sin^2 \theta \):
\[ \sin^2 \theta = 1 - \frac{4}{5} = \frac{1}{5} \]

Finally, take the square root to find \( \sin \theta = \pm \frac{\sqrt{5}}{5} \). This identity not only aids in computing trig functions but also fosters connections between them, reinforcing a deeper understanding of circular and right-angle trigonometry.