Problem 11 - Thomas Calculus

Question

In Exercises \(7-12,\) one of \(\sin x, \cos x,\) and \(\tan x\) is given. Find the other two if \(x\) lies in the specified interval. $$ \tan x=\frac{1}{2}, \quad x \in\left[\pi, \frac{3 \pi}{2}\right] $$

Step-by-Step Solution

Answer

\( \cos x = -\sqrt{\frac{4}{5}}, \sin x = -\sqrt{\frac{1}{5}} \).

Step 1: Determine the Quadrant

Given that \( x \) lies in \( \left[ \pi, \frac{3\pi}{2} \right] \), we know that \( x \) is in the third quadrant. In the third quadrant, both sine and cosine are negative, while tangent is positive.

Step 2: Use Tangent Identity

The given value of tangent is \( \tan x = \frac{1}{2} \). The identity for tangent is \( \tan x = \frac{\sin x}{\cos x} \). Since we know tangent is positive, both sine and cosine must be negative to satisfy the identity in the third quadrant.

Step 3: Find Cosine Using Pythagorean Identity

We use the identity \( 1 + \tan^2 x = \frac{1}{\cos^2 x} \) to find cosine. Substitute \( \tan x = \frac{1}{2} \):\[ 1 + \left( \frac{1}{2} \right)^2 = \frac{1}{\cos^2 x} \]\[ 1 + \frac{1}{4} = \frac{1}{\cos^2 x} \]\[ \frac{5}{4} = \frac{1}{\cos^2 x} \]\[ \cos^2 x = \frac{4}{5} \]Thus, \( \cos x = -\sqrt{\frac{4}{5}} \).

Step 4: Find Sine Using Cosine

We use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find sine. Substitute \( \cos^2 x = \frac{4}{5} \) into the identity:\[ \sin^2 x + \frac{4}{5} = 1 \]\[ \sin^2 x = 1 - \frac{4}{5} \]\[ \sin^2 x = \frac{1}{5} \]Thus, \( \sin x = - \sqrt{\frac{1}{5}} \), since sine is negative in the third quadrant.

Key Concepts

Tangent Function

The tangent function, often abbreviated as \( \tan \), is a fundamental element in trigonometry. It is defined as the ratio of the sine function to the cosine function:\[ \tan x = \frac{\sin x}{\cos x} \]This relationship means that if you know the tangent of an angle, you can often find the sine and cosine values based on their ratios. The tangent function is periodic and repeats every \( \pi \) radians. This means that its values recur after each \( \pi \) interval. The sign of \( \tan x \) depends on the signs of \( \sin x \) and \( \cos x \). For example, in the third quadrant, where the angle \( x \) lies between \( \pi \) and \( \frac{3\pi}{2} \), both sine and cosine are negative, making the tangent function positive.

• If \( \tan x \) is positive, it indicates both \( \sin x \) and \( \cos x \) have the same sign.
• If \( \tan x \) is negative, \( \sin x \) and \( \cos x \) have opposite signs.

The tangent function is unique in its ability to help distinguish the angle's quadrant when its sign is known. Understanding this function's properties aids in solving trigonometric identities and problems effectively.

Sine and Cosine Relationship

Sine and cosine are the foundational trigonometric functions that describe the relationships within a right triangle. These functions also relate to the unit circle. The sine of an angle \( x \), denoted as \( \sin x \), represents the y-coordinate of a point on the unit circle, while the cosine \( \cos x \) represents the x-coordinate.

• Both sine and cosine have a range of values from -1 to 1, and they are periodic with a period of \( 2\pi \).
• Their values change with the angle, following specific patterns as the point moves along the circle.

In the problem where \( \tan x = \frac{1}{2} \), we use the relationship \( \tan x = \frac{\sin x}{\cos x} \) to find \( \sin x \) and \( \cos x \). Given that \( x \) is in the third quadrant, both functions are negative, and this information helps determine the signs of \( \sin x \) and \( \cos x \). This understanding is crucial for working with trigonometric identities and equations.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, relating the squares of the sine and cosine of an angle. The primary form of the identity is:\[ \sin^2 x + \cos^2 x = 1 \]This equation summarizes the idea that a point on the unit circle can be described by its sine and cosine coordinates, and the sum of their squares will always equal one.

• This identity is used to find unknown trigonometric values when one is provided.
• Sub-identities stemming from this include tangent and secant relationships, such as \( 1 + \tan^2 x = \sec^2 x \).

In the given exercise where \( \tan x = \frac{1}{2} \), we employ a modified Pythagorean identity, \( 1 + \tan^2 x = \frac{1}{\cos^2 x} \), to derive the value for cosine. We proceed by calculating \( \sin x \) using \( \sin^2 x = 1 - \cos^2 x \), reinforcing the application of the identities. Understanding how these identities interlink helps solve more complex trigonometric problems and provides insight into the harmonious structure of trigonometric functions.