Problem 67 - Calculus for Biology and Medicine

Question

(a) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \sin \alpha=-\frac{1}{2} \sqrt{3} $$ (b) Find the values of \(\alpha \in[0,2 \pi)\) that satisfy $$ \tan \alpha=\sqrt{3} $$

Step-by-Step Solution

Answer

(a) \(\alpha = 4\pi/3, 5\pi/3\); (b) \(\alpha = \pi/3, 4\pi/3\).

Step 1: Understand the Problem

We need to find solutions for two trigonometric equations. In part (a), we solve for \(\alpha\) in the equation \(\sin \alpha = -\frac{1}{2} \sqrt{3} \). In part (b), we solve for the same angle in the equation \(\tan \alpha = \sqrt{3}\).

Step 2: Recognize Reference Angles

For part (a), we recognize that \(-\frac{1}{2} \sqrt{3}\) is not common, but we know that \(\sqrt{3}/2\) corresponds to \(60^\circ \) (or \(\pi/3\) in radians). Since sine is negative, \(\alpha\) should be in the third or fourth quadrant. For part (b), \(\tan\alpha = \sqrt{3}\) corresponds to the reference angle \(60^\circ\) or \(\pi/3\) in the first quadrant.

Step 3: Find Solutions for Part (a)

The sine function is negative in the third and fourth quadrants. The reference angle for \( \alpha \) is \(\pi/3\): - Third quadrant: \(\alpha = \pi + \pi/3 = 4\pi/3\) - Fourth quadrant: \(\alpha = 2\pi - \pi/3 = 5\pi/3\) Thus, the solutions for part (a) are \(\alpha = 4\pi/3, 5\pi/3\).

Step 4: Find Solutions for Part (b)

The tangent function is positive in the first and third quadrants. The reference angle for \(\alpha\) is \(\pi/3\): - First quadrant: \(\alpha = \pi/3\) - Third quadrant: \(\alpha = \pi + \pi/3 = 4\pi/3\) Thus, the solutions for part (b) are \(\alpha = \pi/3, 4\pi/3\).

Step 5: Combine Results

We have found that:- For part (a), \(\alpha = 4\pi/3, 5\pi/3\).- For part (b), \(\alpha = \pi/3, 4\pi/3\).

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions involving trigonometric functions such as sine, cosine, and tangent. They equate a trigonometric function to a specific value and ask us to find the angles that satisfy this equality. For instance, in the exercise provided, we have two equations: \( \sin \alpha = -\frac{1}{2} \sqrt{3} \) and \( \tan \alpha = \sqrt{3} \). The goal is to find all the angles \( \alpha \) within the interval \([0, 2\pi)\) that make these equations true.

To solve such equations, we typically follow a process:

• Identify the reference angle from the trigonometric value given.
• Determine the quadrants where the function (sine, tangent) is positive or negative.
• Use these quadrants to find all possible angles that satisfy the original equation.

This process simplifies solving the equations as it breaks down the task into systematic steps. Understanding trigonometric equations is crucial for various fields such as physics, engineering, and computer science.

Reference Angles

A reference angle is the acute angle that a given angle forms with the x-axis. This concept is crucial when solving trigonometric equations because it helps determine the possible solutions by manipulating standard angles. When encountering a trigonometric value, determining the reference angle is often the first step towards finding all that satisfy a particular equation.

For example, in the equation \( \tan \alpha = \sqrt{3} \), the reference angle is \( 60^\circ \) or \( \pi/3 \) radians. Recognizing this reference angle helps us not only identify the possible angles in standard position but also helps us use symmetry to find other solutions in different quadrants. Reference angles enable us to use known angle values to solve equations quickly, sidestepping the arduous task of memorizing long lists of trigonometric values.

Quadrants

The concept of quadrants is fundamental in trigonometry. The coordinate plane is divided into four quadrants, each with its own characteristics regarding the sign of the trigonometric functions. Knowing these characteristics is essential in solving trigonometric equations.

Let's break down the quadrants:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, but cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, and sine and tangent are negative.

Understanding which quadrant an angle lies in tells us whether sine, cosine, or tangent values are positive or negative. This information is crucial when you have equations where the function values can be negative or positive depending on the equation. For example, knowing that sine is negative in the third and fourth quadrants helped in identifying the possible solutions for part (a). Similarly, recognizing that tangent is positive in the first and third quadrants guided us in solving part (b) of the original exercise. By using quadrants, we can efficiently find solutions without unnecessary complexity.