Problem 68 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

(a) \( \alpha = \frac{3\pi}{4}, \frac{5\pi}{4} \); (b) \( \alpha = \frac{\pi}{3}, \frac{5\pi}{3} \).

Step 1: Understanding the Problem for Part (a)

We need to determine which angles \( \alpha \) within the interval \([0, 2\pi)\) result in \( \cos \alpha = -\frac{1}{2} \sqrt{2} \). This requires us to recall the cosine function's properties and how it relates to angles in different quadrants.

Step 2: Analyzing Cosine Function for Part (a)

The cosine function is negative in the second and third quadrants. We rewrite the given equation as \( \cos \alpha = -\frac{\sqrt{2}}{2} \). We recall that \( \cos \alpha = -\frac{\sqrt{2}}{2} \) at angles \( \alpha = \frac{3\pi}{4} \) and \( \alpha = \frac{5\pi}{4} \).

Step 3: Concluding Part (a)

Thus, the values of \( \alpha \) that satisfy \( \cos \alpha = -\frac{1}{2} \sqrt{2} \) are \( \alpha = \frac{3\pi}{4} \) and \( \alpha = \frac{5\pi}{4} \). These values lie between \([0, 2\pi)\).

Step 4: Understanding the Problem for Part (b)

Next, we need to find the angles \( \alpha \) within \([0, 2\pi)\) that satisfy \( \sec \alpha = 2 \). The secant function is the reciprocal of the cosine function, so this implies that \( \cos \alpha = \frac{1}{2} \).

Step 5: Analyzing Cosine Function for Part (b)

The cosine function \( \cos \alpha = \frac{1}{2} \) at angles in the first and fourth quadrants. Specifically, \( \alpha = \frac{\pi}{3} \) and \( \alpha = \frac{5\pi}{3} \) are the angles between \([0, 2\pi)\) where the cosine value is \( \frac{1}{2} \).

Step 6: Concluding Part (b)

Thus, the values of \( \alpha \) that satisfy \( \sec \alpha = 2 \) are \( \alpha = \frac{\pi}{3} \) and \( \alpha = \frac{5\pi}{3} \).

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry. It helps determine the relationship between the angles and sides of right-angled triangles. In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. This function is periodic and repeats every \(2\pi\) radians.

• The cosine function varies between -1 and 1.
• It is positive in the first and fourth quadrants.
• It is negative in the second and third quadrants.

In the standard angle measurement system, such as the interval from 0 to \(2\pi\), knowing this sign change is crucial. For solutions to equations like \(\cos \alpha = -\frac{1}{2}\sqrt{2}\), we have to focus on the quadrants where cosine values are negative, which are the second and third quadrants. Hence, we find specific angles, for example, \(\alpha = \frac{3\pi}{4}\) and \(\alpha = \frac{5\pi}{4}\). Understanding the cosine function's behavior is key to solving trigonometric equations.

Secant Function

The secant function (sec) is less commonly discussed, but crucial for rounding out our understanding of trigonometric functions. Secant is defined as the reciprocal of the cosine function. Mathematically, this means \({\text{sec}\, \alpha = \frac{1}{\cos \alpha}}\).

• When \(\cos \alpha = 0\), \(\sec \alpha\) is undefined.
• Secant shares many periodic properties with cosine because it is derived from it, repeating every \(2\pi\) radians.

For the equation \(\sec \alpha = 2\), it's essentially asking us to find when \(\cos \alpha = \frac{1}{2}\). Since cosine \(\alpha\) returns \(\frac{1}{2}\) in the first and fourth quadrants, it results in angles \(\alpha = \frac{\pi}{3}\) and \(\alpha = \frac{5\pi}{3}\). Understanding these relationships and how the secant behaves can simplify solving trig equations.

Unit Circle

The unit circle is a significant tool in trigonometry, providing a geometric perspective for trigonometric functions. It's a circle with a radius of 1, centered at the origin of the coordinate plane. Each point on the unit circle represents a cosine and sine value for a corresponding angle.

• Cosine of an angle is the x-coordinate on the unit circle.
• Sine of an angle is the y-coordinate on the unit circle.

The unit circle helps visualize solutions to trigonometric equations. For example, angles that correspond to a negative cosine value, like \(\cos \alpha = -\frac{1}{2}\sqrt{2}\), visually appear in the second and third quadrants of the circle. The unit circle's simplicity and consistency help us reliably interpret trigonometric functions, identifying when the cosine or secant will yield given values.

Quadrants

The concept of quadrants is essential for understanding angles and trigonometric functions. A plane is divided into four quadrants, using the x-axis and y-axis as reference lines, each quadrant representing 90 degrees, or \(\frac{\pi}{2}\) radians. These quadrants influence the signs of sine, cosine, and other trigonometric functions.

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive, cosine and tangent are negative.
• Third Quadrant: Tangent is positive, sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, sine and tangent are negative.

Understanding where each function is positive or negative helps determine the solutions for trigonometric equations. For instance, knowing that cosine is negative in the second and third quadrants narrows down the possibilities for angle solutions within given intervals. It's vital to grasp quadrant behaviors to solve problems like finding specific \(\alpha\) values that satisfy trigonometric equations.