Problem 43 - Calculus: One and Several Variables

Question

Find the number \((s) x\) in the interval \([0,2 \pi j]\) which satisfy the equation. $$\tan (x / 2)=-1$$

Step-by-Step Solution

Answer

The solution to the equation \(\tan (x / 2)=-1\) within the interval \([0,2 \pi]\) is \(x = \pi\).

Step 1: Recognize the properties of the tangent function

Particularly important here are the following two properties: a) The tangent function \(\tan(\theta)\) has roots at \(\theta = n \pi\), where \(n\) is an integer. b) \(\tan(\theta)\) is negative in the second and fourth quadrants of the unit circle.

Step 2: Set up the equation and solve the inner function

The given equation is \(\tan(x / 2)=-1\). Therefore, the angle representation for \(\tan(\theta)\) to be -1 is \(\theta = \pi + n \pi\), where \(n\) is an integer.

Step 3: Solve for \(x\)

The inner function in the equation is \(x/2\). Therefore, we set \(x/2 = \pi + n \pi\) and solve for \(x\) to get \(x = 2\pi + 2n\pi\)

Step 4: List the solutions within the interval \([0,2 \pi]\)

As \(x\) should be between 0 and \(2 \pi\), we must find all the \(x\) values within this interval. Plugging in values of \(n\) that satisfy the condition, we find \(x = \pi\) when \(n=0\), and \(x = 2\pi\) when \(n=1\). Note that \(x = 2\pi\) should be excluded as our interval is closed at 0 but open at \(2\pi\).

Key Concepts

Tangent Function

The tangent function is a critical component of trigonometry and is often used to solve various types of equations involving angles. The tangent of an angle, denoted as \(\tan(\theta)\), is defined as the ratio between the opposite side and the adjacent side in a right-angled triangle. It is one of the primary trigonometric functions alongside sine and cosine. Here are some key properties to remember:

• The function repeats every \(\pi\) radians, meaning it is periodic with a period of \(\pi\).
• Tangent exhibits vertical asymptotes where the function is undefined, specifically when the angle \(\theta\) is \((\frac{\pi}{2} + n\pi)\), \(n\) being an integer.
• The tangent function is positive when angles are in the first and third quadrants, and it becomes negative in the second and fourth quadrants of the unit circle.

Understanding these characteristics of the tangent function is pivotal when solving equations like \(\tan(x/2) = -1\). Recognizing where the tangent function is negative helps in determining the correct angles that satisfy given conditions.

Unit Circle

The unit circle is a crucial tool in trigonometry that helps to visualize and solve trigonometric equations. It is a circle with a radius of one unit and is centered at the origin of a coordinate plane. Each point on the unit circle corresponds to a specific angle measured from the positive x-axis. One full rotation around the circle represents an angle of \(2\pi\) radians, or 360 degrees.When considering the tangent function on the unit circle, it's important to recognize that:

• Angles in the first quadrant (0 to \(\frac{\pi}{2}\)) have positive tangent values.
• In the second quadrant (\(\frac{\pi}{2}\) to \(\pi\)), the tangent values are negative.
• Angles in the third quadrant (\(\pi\) to \(\frac{3\pi}{2}\)) yield positive tangent values.
• In the fourth quadrant (\(\frac{3\pi}{2}\) to \(2\pi\)), the tangent values become negative again.

When solving an equation like \(\tan(x/2)=-1\), it is the negative value of the tangent function that indicates the solution lies in either the second or fourth quadrant. Thus, the unit circle helps in strategizing which angles are potential candidates for solutions within a given interval.

Interval Solutions

Solving trigonometric equations within a specified interval is a common task in mathematics. It requires understanding how solutions can repeat due to the periodic nature of trigonometric functions. When tackling equations like \(\tan(x/2) = -1\), solutions are derived for \(x/2\) based on the properties of tangent. Because tangent repeats every \(\pi\), solutions are often represented in terms of \(\pi + n\pi\), where \(n\) is an integer.For the specific interval \([0, 2\pi)\):

• The general solution for \(x/2\) is \(\pi + n\pi\), leading to the equation for \(x\) as \(2\pi + 2n\pi\).
• We then find values of \(x\) within the interval by substituting different integer values for \(n\), paying attention to the range restriction. For the interval \([0, 2\pi)\), \(x = \pi\) fits perfectly while \(x = 2\pi\) does not, since the interval is open at \(2\pi\).

Understanding interval solutions is paramount to accurately solving trigonometric equations, as it ensures solutions are confined to a realistic and specified range. This approach is not only essential for precise solutions but also for comprehending how periodic behavior influences available solutions.