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

\( \sin x = -\frac{\sqrt{5}}{5}, \; \cos x = -\frac{2\sqrt{5}}{5} \).

Step 1: Determine the quadrant

The interval given is \( x \in \left[ \pi, \frac{3\pi}{2} \right] \), which lies in the third quadrant. In this quadrant, sine and cosine are both negative, and tangent is positive.

Step 2: Use Pythagorean identity for tangent

The identity \( 1 + \tan^2 x = \sec^2 x \) can be used to relate tangent and secant. Here, \( \tan x = \frac{1}{2} \), so substitute and solve: \( 1 + \left(\frac{1}{2}\right)^2 = \sec^2 x \). This simplifies to \( \sec^2 x = \frac{5}{4} \), thus \( \sec x = \pm \frac{\sqrt{5}}{2} \). Since cosine is negative in the third quadrant, \( \sec x = -\frac{\sqrt{5}}{2} \).

Step 3: Find cosine using secant

Remember, \( \sec x = \frac{1}{\cos x} \). Solve \( \cos x = -\frac{2}{\sqrt{5}} \), which can be rationalized to \( \cos x = -\frac{2\sqrt{5}}{5} \).

Step 4: Use Pythagorean identity for cosine

Use the identity \( \sin^2 x + \cos^2 x = 1 \) to find sine. Substitute \( \cos x = -\frac{2\sqrt{5}}{5} \) into the equation: \( \sin^2 x + \left(-\frac{2\sqrt{5}}{5}\right)^2 = 1 \). Simplifying, you get \( \sin^2 x = \frac{1}{5} \).

Step 5: Solve for sine

Calculate \( \sin x = \pm \frac{1}{\sqrt{5}} \). Since sine is negative in the third quadrant, \( \sin x = -\frac{1}{\sqrt{5}} \), which can be rationalized to \( \sin x = -\frac{\sqrt{5}}{5} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are the cornerstone of trigonometry, and they describe relationships in right-angled triangles. These functions are crucial for solving problems involving angles and lengths. The three primary trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)).

• Sine: For a given angle in a right triangle, sine is the ratio of the length of the opposite side to the hypotenuse.
• Cosine: Cosine is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent: Tangent is the ratio of the opposite side to the adjacent side.

In addition to these, there are three reciprocal functions: cosecant (\( \csc \theta = \frac{1}{\sin \theta} \)), secant (\( \sec \theta = \frac{1}{\cos \theta} \)), and cotangent (\( \cot \theta = \frac{1}{\tan \theta} \)). Understanding these relationships is key when working with angles and solving trigonometric identities.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, each affecting the sign of trigonometric functions. Knowing the quadrant in which an angle resides helps determine the positivity or negativity of the trigonometric functions.

• First Quadrant: Angles between 0 and \( \frac{\pi}{2} \). Here, all trigonometric functions are positive.
• Second Quadrant: Angles between \( \frac{\pi}{2} \) and \( \pi \). Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Angles between \( \pi \) and \( \frac{3\pi}{2} \). Both sine and cosine are negative, but tangent is positive.
• Fourth Quadrant: Angles between \( \frac{3\pi}{2} \) and \( 2\pi \). Cosine is positive, with sine and tangent being negative.

In the given exercise, angles are in the third quadrant, thus impacting the signs of sine, cosine, and tangent functions directly. Being familiar with these quadrant properties helps solve trigonometric problems accurately.

Pythagorean Identities

Pythagorean identities are equations that express the intrinsic relationships between the sine, cosine, and tangent functions. Named after the Pythagorean theorem, they are critical in simplifying expressions and solving trigonometric equations.

Basic Pythagorean Identity

The most fundamental identity is \( \sin^2 x + \cos^2 x = 1 \). This equation is derived from the Pythagorean theorem and is used frequently to find missing sine or cosine values when one is known.

Secant and Tangent Relationship

Another important identity is \( 1 + \tan^2 x = \sec^2 x \), which is used to find relationships involving tangent and secant. For instance, given \( \tan x \) in a problem, you can use this identity to determine \( \sec x \), as in the given exercise.

Cosecant and Cotangent Relationship

Additionally, \( 1 + \cot^2 x = \csc^2 x \) describes relationships for cotangent and cosecant. These identities are essential tools in trigonometry, enabling students to handle a wide variety of problems involving angles, and solve complex trigonometric equations efficiently.