Problem 57 - Precalculus: Functions and Graphs

Question

Graphically solve the trigonometric equation on the indicated interval to two decimal places. \(\tan \left(\frac{1}{2} x+1\right)=\sin \frac{1}{2} x ; \quad[-2 \pi, 2 \pi]\)

Step-by-Step Solution

Answer

The solutions are approximately \(-5.46, -1.57, 2.32,\) and \(5.18\).

Step 1: Understand the Equation

The given equation is \(\tan\left(\frac{1}{2}x + 1\right) = \sin\frac{1}{2}x\). We need to solve this equation graphically over the interval \([-2\pi, 2\pi]\).

Step 2: Analyze the Functions

Identify the functions involved: \(f(x) = \tan\left(\frac{1}{2}x + 1\right)\) and \(g(x) = \sin\frac{1}{2}x\). We need to plot these two functions to find their intersections, which represent solutions to the equation.

Step 3: Plot the Functions

Graph \(f(x) = \tan\left(\frac{1}{2}x + 1\right)\) and \(g(x) = \sin\frac{1}{2}x\) over the interval \([-2\pi, 2\pi]\). Use a graphing calculator or software to visualize the graphs. Look carefully for points where both graphs intersect.

Step 4: Identify Intersections

Locate the points where the graphs intersect within the interval \([-2\pi, 2\pi]\). These intersections are the solutions to the equation. Make sure to find the x-values of these points to two decimal places.

Step 5: Verify and List Solutions

Check the x-values from the intersections to ensure they satisfy the initial equation. List all valid solutions within the specified interval to two decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are used to describe the relationships between the angles and sides of triangles. They are periodic and include functions like sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each of these functions has unique properties:

• Sine (\(\sin\)): This function relates the angle of a right triangle to the ratio of the length of the opposite side and the hypotenuse. It is bounded between -1 and 1.
• Tangent (\(\tan\)): This trigonometric function is the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with period \(\pi\) and has vertical asymptotes where it is undefined.

For the given exercise, identifying and differentiating these functions is crucial since \(\tan\left(\frac{1}{2}x + 1\right)\) and \(\sin\left(\frac{1}{2}x\right)\) play essential roles in solving the equation. Understanding their behavior and characteristics helps us interpret their graphical intersections meaningfully.

Graphing Techniques

Graphing techniques involve the use of graphing tools, such as graphing calculators or software, to visualize functions. When solving trigonometric equations graphically, it is vital to accurately plot the functions over the specified interval.
For our exercise, we need to visualize \(f(x) = \tan\left(\frac{1}{2}x + 1\right)\) and \(g(x) = \sin\left(\frac{1}{2}x\right)\):

• Set the Scale: Start by setting up your graph with the x-axis ranging from \([-2\pi, 2\pi]\) and choosing an appropriate scale to capture the oscillations of the trigonometric functions.
• Plot the Functions: Use graphing software to input the equations correctly. This step might involve recognizing how transformations affect their graphs, such as horizontal shifts or stretches.

Accurate plotting will help you identify solutions as intersection points within the interval, making this a crucial step in the graphical solution process.

Intersection Points

Intersection points on the graph represent solutions to the trigonometric equation because they indicate where the two functions, \(f(x)\) and \(g(x),\) have the same y-value for a given x-value. Finding these points is the key to solving the equation graphically. It can be done by inspecting the graph visually or using technological tools to pinpoint exact intersection coordinates.
Here are some tips:

• Careful Observation: Look closely at where the graphs of \(\tan\left(\frac{1}{2}x + 1\right)\) and \(\sin\left(\frac{1}{2}x\right)\) intersect. These are your potential solutions.
• Precision Tools: Graphing calculators with intersection-finding functions can offer precise calculations, often to two or more decimal places, highlighting exact x-values where intersections occur.

Verifying that these points lie within the specified interval, \([-2\pi, 2\pi]\) is necessary for confirming the validity of the solutions.

Interval Analysis

Interval analysis involves examining the given range of values within which we are interested in finding solutions. In this trigonometric equation, the specified interval is \([-2\pi, 2\pi]\), which includes both positive and negative values of \(x\).
When performing interval analysis:

• Respect the Bounds: Ensure that all potential solutions (intersection points) adhere to this interval. Any solutions found outside must be disregarded.
• Symmetry Consideration: Understand that due to the periodic nature of trigonometric functions, solutions might repeat. However, inclusion in the specified interval is necessary for them to be valid.

By thoroughly analyzing the interval and cross-referencing with the graph, you confirm the acceptable \(x\)-values that solve the equation accurately.