Problem 27 - Precalculus with Calculus Previews

Question

If \(3 \cos x=\sin x\), find all values of \(\tan x, \cot x\) \(\sec x,\) and \(\csc x\)

Step-by-Step Solution

Answer

\( \tan x = 3, \cot x = \frac{1}{3}, \sec x = \sqrt{10}, \csc x = \frac{\sqrt{10}}{3} \).

Step 1: Establish a Relationship

We start with the equation given: \( 3\cos x = \sin x \). Divide both sides by \( \cos x \) to express the relationship in terms of \( \tan x \): \( 3 = \tan x \), since \( \tan x = \frac{\sin x}{\cos x} \).

Step 2: Compute Tangent and Cotangent

From Step 1, we found that \( \tan x = 3 \). The cotangent is the reciprocal of the tangent, so \( \cot x = \frac{1}{\tan x} = \frac{1}{3} \).

Step 3: Use Pythagorean Identity to Find Cosine and Sine

We use the identity \( 1 + \tan^2 x = \sec^2 x \). Substitute \( \tan x = 3 \) into the equation: \( 1 + 3^2 = \sec^2 x \) which simplifies to \( 1 + 9 = \sec^2 x \). Therefore, \( \sec^2 x = 10 \).

Step 4: Compute Secant and Cosecant

From Step 3, \( \sec x = \sqrt{10} \). The secant is the reciprocal of the cosine, \( \sec x = \frac{1}{\cos x} \), so \( \cos x = \frac{1}{\sqrt{10}} \). Next, find \( \sin x = \tan x \cdot \cos x = 3 \cdot \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}} \). The cosecant is the reciprocal of the sine: \( \csc x = \frac{1}{\sin x} = \frac{\sqrt{10}}{3} \).

Key Concepts

Tangent Function

The tangent function, represented as \( \tan x \), is a fundamental trigonometric function that arises from dividing sine by cosine. In mathematical terms, it is expressed as:

• \( \tan x = \frac{\sin x}{\cos x} \)

When given an equation like \( 3\cos x = \sin x \), one can simplify it to find the tangent of \( x \) by dividing both sides by \( \cos x \). This gives a clearer relationship and allows us to express tangent directly:
- In this case, \( \tan x = 3 \).
An important aspect to remember about tangent is that it's undefined wherever cosine is zero, as division by zero is not possible. This makes tangent periodic and undefined at certain angles, such as 90° (or multiples of \( \pi/2 \) radians).
To find the cotangent, which is the reciprocal of the tangent, use:

• \( \cot x = \frac{1}{\tan x} \)

With the given \( \tan x = 3 \), it follows that \( \cot x = \frac{1}{3} \). Understanding these concepts can help in solving trigonometric equations and analyzing functions.

Reciprocal Identities

Reciprocal identities are vital in understanding the relationships between trigonometric functions. They provide a straightforward way to switch between functions and their reciprocals. Let's look at some key reciprocal identities:

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

These identities are handy in cases where you need to convert one trigonometric function into another or solve equations that involve different trigonometric functions.
From the example, we start by recognizing that \( \tan x = 3 \), giving us \( \cot x = \frac{1}{3} \) as already found.
Moving next to \( \sec x \), since it's the reciprocal of \( \cos x \), we determine it as \( \sqrt{10} \) from the Pythagorean identity calculations. Similarly, using \( \sin x = \frac{3}{\sqrt{10}} \), the reciprocal identity tells us \( \csc x = \frac{\sqrt{10}}{3} \).
These identities simplify the complex relationships within trigonometric expressions and give alternate ways to express angles and functions.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry linking sine, cosine, and tangent. It's expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)
• Another useful form is \( 1 + \tan^2 x = \sec^2 x \)

These identities are called 'Pythagorean' because they resemble the Pythagorean theorem from geometry \((a^2 + b^2 = c^2)\).
In trigonometry, these identities help solve equations by linking different functions. For example, to find \( \sec x \) when \( \tan x = 3 \), the identity \( 1 + \tan^2 x = \sec^2 x \) enables us to find \( \sec^2 x = 10 \).
From there, taking the square root aids in determining \( \sec x = \sqrt{10} \). This process bridges the relationship between tangent and secant through the identity.
Understanding these identities allows students to simplify complex trigonometric expressions and solve a system of equations concerning an angle, as seen in this exercise.