Problem 21 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 3 \tan \theta-1=\tan \theta $$

Step-by-Step Solution

Answer

The solutions for \(\theta\) are \(\pi/6\) and \(7\pi/6\)

Step 1: Isolate terms

First, let's isolate \(\tan \theta\) on one side by subtracting \(\tan \theta\) from both sides of the equation. The equation becomes: \( 3\tan \theta - \tan \theta = 1 \) which simplifies to \(2\tan \theta = 1\).

Step 2: Solve for \(\tan \theta\)

Next, let's solve for \(\tan \theta\) by dividing both sides of the equation by 2. This will give: \(\tan \theta = 1/2\).

Step 3: Solve for \(\theta\)

Now, we determine the value of \(\theta\). Remembering the range of \(\theta\) (from step 1), we can say \(\theta = \arctan(1/2)\). This gives \(\theta = \pi/6\) or \(\theta = 7\pi/6\).

Step 4: Verifying solution

Finally, we verify that the solutions fall within the given range \(0 \leq \theta < 2 \pi \). Indeed, both \(\pi/6\) and \(7\pi/6\ falls\) within this range.

Key Concepts

Tangent function

The tangent function, denoted as \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the sine to the cosine of an angle \( \theta \). Therefore, we have:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

This function is periodic, with a period of \( \pi \). This means that if \( \tan \theta = x \), it will also hold that \( \tan(\theta + n\pi) = x \) for any integer \( n \).
The tangent function is undefined at angles where \( \cos \theta = 0 \), which happen at \( \frac{\pi}{2} + n\pi \). It is important in solving trigonometric equations as it helps relate angles in a right triangle to the lengths of its sides.

Angle measurement in radians

Radians are a unit of angular measurement that is based on the radius of a circle. They provide a natural way to describe angles because the angle in radians is the ratio of the arc length to the radius of the circle. Consider the following:

• \( 2\pi \) radians is equivalent to 360 degrees.
• Thus, \( \pi \) radians equals 180 degrees, and \( \frac{\pi}{2} \) radians equals 90 degrees.

Using radians simplifies many mathematical problems, especially those involving calculus and trigonometry, due to its direct relationship with the unit circle. In trigonometry, we often solve problems in the interval \( [0, 2\pi) \), meaning from \( 0 \) to just under \( 2\pi \). This is because \( 2\pi \) represents a full rotation around a circle.

Solving trigonometric equations

Solving trigonometric equations involves finding angles that make the equation true. The process often involves isolating the trigonometric function and using inverse functions to find specific angle values. Here are the key steps typically involved:

• Rearranging the equation to isolate the trigonometric function on one side.
• Using inverse trigonometric functions, such as \( \arctan \), \( \arcsin \), or \( \arccos \), to find the angle \( \theta \).
• Considering the function’s periodicity to find all possible solutions within the specified range, like \( [0, 2\pi) \).

In our problem, \( 3\tan \theta - 1 = \tan \theta \) was simplified to \( 2\tan \theta = 1 \), leading to \( \tan \theta = \frac{1}{2} \).
Using the inverse function \( \theta = \arctan(\frac{1}{2}) \), we found specific solutions \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{7\pi}{6} \) within the specified range for \( \theta \).

Unit circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of the coordinate system. The unit circle helps in understanding the values of trigonometric functions and their behavior. Key points on the unit circle include:

• The x-coordinate of a point on the unit circle corresponds to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.
• The tangent of the angle is represented by the ratio of the y-coordinate to the x-coordinate, which is \( \tan \theta = \frac{y}{x} \).

Thus, every angle \( \theta \) can be associated with a point on the unit circle. This makes it easy to visualize and understand angles in radians and their corresponding trigonometric function values. For \( \tan \theta = \frac{1}{2} \), one would look for angles \( \theta \) where this ratio holds true, as found at angles \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{7\pi}{6} \).
The unit circle is crucial in visualizing these solutions, ensuring they fall within the desired range of \( 0 \leq \theta < 2\pi \).