Problem 160 - Calculus Volume 1

Question

Solve the trigonometric equations on the interval \(0 \leq \theta<2 \pi\). \(3 \sec \theta-2 \sqrt{3}=0\)

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{6}, \frac{11\pi}{6}\)

Step 1: Rewrite the equation in terms of cosine

The equation given is \(3 \sec \theta - 2 \sqrt{3} = 0\). The secant function is the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\). Substitute this into the equation: \(3 \left(\frac{1}{\cos \theta}\right) = 2 \sqrt{3}\).

Step 2: Solve for cosine theta

Continue solving the equation: \(\frac{3}{\cos \theta} = 2 \sqrt{3}\). Multiply both sides by \(\cos \theta\), to get \(3 = 2 \sqrt{3} \cdot \cos \theta\). Divide both sides by \(2 \sqrt{3}\) to isolate \(\cos \theta\): \(\cos \theta = \frac{3}{2 \sqrt{3}}\). Simplify the expression: \(\cos \theta = \frac{\sqrt{3}}{2}\).

Step 3: Find theta using the unit circle

The cosine of an angle is \(\frac{\sqrt{3}}{2}\) at angles \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\) on the interval \(0 \leq \theta < 2\pi\). These angles correspond to the positions on the unit circle where the x-coordinate (cosine value) is \(\frac{\sqrt{3}}{2}\).

Step 4: Write the final solution

The solutions to the trigonometric equation are \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\). These are the values of \(\theta\) that satisfy \(3 \sec \theta - 2 \sqrt{3} = 0\) on the interval \(0 \leq \theta < 2\pi\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle allows us to define trigonometric functions for all angles.
Here’s how it works:

• The angle \(\theta\) is measured from the positive x-axis.
• The coordinates of a point on this circle are \((\cos \theta, \sin \theta)\).
• The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine.

Angles increase in a counter-clockwise direction. Starting from the positive x-axis, right side of the circle, you move around to define different trigonometric function values.
Knowing key angles on the unit circle helps solve trigonometric equations efficiently. For example, the x-coordinate, or cosine value, for \(\theta = \frac{\pi}{6}\) is \(\frac{\sqrt{3}}{2}\). Being able to associate these angles with their cosine values allows you to easily find solutions to trigonometric equations.

Secant Function

The secant function, denoted as \(\sec \theta\), is one of the reciprocal trigonometric functions. It represents the reciprocal of the cosine function. So, the formula is \(\sec \theta = \frac{1}{\cos \theta}\).

Using the secant function can be helpful especially when solving equations that involve division by cosine. Here are some key points:

• The secant function is undefined whenever the cosine function is zero.
• In terms of the unit circle, secant can be thought of as stretching or compressing the x-coordinate (cosine value).
• For angles where cosine is greater than 1 or less than -1, secant takes on very large values because of this reciprocal relationship.

While not as commonly used as sine and cosine, the secant function is critical in trigonometry for understanding these reciprocal relationships, providing an alternative method of representation.

Cosine Function

Among the three fundamental trigonometric functions, the cosine function is perhaps the most relatable when connected to real-world applications. Cosine, denoted as \(\cos \theta\), gives us the horizontal component of an angle on the unit circle.

Understanding cosine is essential because:

• Cosine values range from -1 to 1, representing the farthest left and right points on the unit circle.
• It's often used to describe wave patterns, such as sound or light.
• Like the sine function, cosine exhibits periodic behavior, repeating every \(2\pi\) radians.

The cosine function is integral to solving trigonometric equations. In our exercise, recognizing the x-coordinate—\(\frac{\sqrt{3}}{2}\)—directs us to specific angles on the unit circle. Cosine helps in finding these angles, particularly when analyzing oscillations or circular movement.