Problem 60 - Precalculus: Functions and Graphs

Question

Graphically solve the trigonometric equation on the indicated interval to two decimal places. \(3 \sin (2 x)+0.5=2 \sin \left(\frac{1}{2} x+1\right) ; \quad[-\pi, \pi]\)

Step-by-Step Solution

Answer

The solutions are approximately found at the intersection points of the graphs over the interval \([-\pi, \pi]\).

Step 1: Understand the Equation

We need to solve the trigonometric equation \(3 \sin(2x) + 0.5 = 2 \sin\left(\frac{1}{2}x + 1\right)\). This involves finding the points where the two sides of the equation (\(3 \sin(2x) + 0.5\) and \(2 \sin\left(\frac{1}{2}x + 1\right)\)) are equal. We will graph each side separately to find these intersection points.

Step 2: Set Up the Graphs

Graph the functions \(y_1 = 3 \sin(2x) + 0.5\) and \(y_2 = 2 \sin\left(\frac{1}{2}x + 1\right)\) on the interval \([-\pi, \pi]\). Use an appropriately detailed scale to accommodate the range and period of these sine functions.

Step 3: Plot Graphs of Both Functions

Using a graphing calculator or graphing software, plot both functions over the interval \([-\pi, \pi]\). Make sure the graph ties in the amplitude, frequency, and phase shift of each sinusoid accurately.

Step 4: Identify Intersections

Look for points where the two graphs intersect within the interval \([-\pi, \pi]\). These intersection points represent the solutions to the original equation.

Step 5: Approximate Solutions to Two Decimal Places

Using the zoom-in feature or cross-referencing with calculated values, identify the \(x\)-coordinates of these intersection points. Round these \(x\)-coordinates to two decimal places.

Key Concepts

graphical solutions

In the realm of trigonometric equations, graphical solutions can be a powerful tool. When faced with an equation like \(3 \sin(2x) + 0.5 = 2 \sin\left(\frac{1}{2}x + 1\right)\), one approach is to visualize each side of the equation graphically. This is done by plotting each trigonometric function on the same set of axes over a defined interval, in this case, \([-\pi, \pi]\).
Seeing these graphs can help you directly identify the solutions as the points where the graphs intersect. Using graphical methods means:

• No need to algebraically manipulate complex trigonometric identities or expressions.
• Immediate visual insight into the behavior of the functions, providing both concept clarity and solution verification.

Graphing tools and calculators are especially useful here, allowing you to "see" the solution and confirm your work.

intersection points

Intersection points in a graphical solution are crucial. They show where two different functions have the same value at the same \(x\). For the equation \(3 \sin(2x) + 0.5 = 2 \sin\left(\frac{1}{2}x + 1\right)\), finding these points within the interval \([-\pi, \pi]\) provides us with the solutions to the equation.
At each intersection point:

• The value of \(y\) is the same for both functions.
• The corresponding \(x\)-coordinate represents a solution to the equation.

Using graphical software or a calculator, you can zoom in to find these points with greater precision, often requiring adjustments to the viewing window to get up close enough to read accurate values. This helps ensure the solutions are precise, as some intersections can appear very close together or at awkward intervals.

sine function

The sine function is one of the foundational trigonometric functions, essential in solving equations like the one encountered. It has the form \(a \sin(bx + c) + d\), where:

• \(a\) affects the amplitude (height of the wave).
• \(b\) affects the frequency (how many waves appear over a particular interval).
• \(c\) provides a horizontal shift (moving the wave left or right).
• \(d\) affects the vertical shift (moving the wave up or down).

In the equation above, both \(3 \sin(2x) + 0.5\) and \(2 \sin\left(\frac{1}{2}x + 1\right)\) are variations of the basic sine function, each with distinct parameters that alter their appearance. Understanding how each parameter alters the sine function's graph is crucial for predicting intersection points and comprehending the equation's behavior over the given interval.

trigonometric functions

Trigonometric functions consist of sine, cosine, and tangent functions among others, each playing pivotal roles in mathematics and applied sciences. These functions exhibit periodic behavior, meaning they repeat at regular intervals, which makes them vital in modeling cyclical phenomena.
In our equation, we deal exclusively with the sine function, a periodic function with a neat, wave-like appearance. Trigonometric functions like sine:

• Are defined for all real numbers.
• Have periods, the interval over which they complete one full cycle of their wave.
• Are used to model phenomena such as sound waves, light cycles, and other sinusoidal patterns.

When multiple trigonometric functions are involved, as in this equation, their intersections indicate key points of equal output, often corresponding to real-world occurrences where cyclic behaviors align. Understanding how to graph and interpret these functions equips us with tools to solve complex trigonometric equations graphically.