Problem 100 - Precalculus Essentials

Question

In Exercises \(99-101,\) graph each equation in \(a\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by [-3,3,1] viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent. $$y=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}$$

Step-by-Step Solution

Answer

The graph represents a standard sine wave which can be represented by \(y = sin(x)\). The two equations \(y=\frac{2 \tan \frac{x}{2}}{1+\tan^{2} \frac{x}{2}}\) and \(y = sin(x)\) are equivalent as they can be transformed into one another using trigonometric double angle identities.

Step 1 - Analyze and Graph the Equation

Analyzing the given equation \(y=\frac{2 \tan \frac{x}{2}}{1+\tan^{2} \frac{x}{2}}\), it can be recognized as the formula for double angle identity in trigonometry. This particular equation given is the formula for sin(x). So, let's use a suitable graphing tool or software to draw the graph, using the given viewing rectangular parameters [-2π, 2π, π/2] by [-3,3,1].

Step 2 - Describe the Graph in Another Equation

Observing the graph, it's identified that it has the pattern of a sine function. It reaches the peak at π/2 in y-direction and has a period of 2π. The equation describing the graph is therefore: \(y = sin(x)\).

Step 3 - Verify the Equivalence of the Two Equations

To verify that the original and the new equation are equivalent, we can use the double angle identities. Double angle formula for sin(x) is \(sin(x) = \frac{2 \tan \frac{x}{2}}{1+\tan^{2} \frac{x}{2}}\). Therefore, these two equations are indeed equivalent.

Key Concepts

Double Angle Identities

Understanding double angle identities is crucial for simplifying and solving trigonometric expressions and equations. These identities express trigonometric functions of double angles in terms of single angles. One of the most commonly used identities is the sine double angle identity, which is given by:
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]
However, there are scenarios where we use a variation of this identity to express \(\sin(\theta)\) in terms of \(\tan(\frac{\theta}{2})\). This is particularly useful when dealing with integrals or equations that involve tangents. The modified identity is:
\[ \sin(\theta) = \frac{2\tan(\frac{\theta}{2})}{1+\tan^{2}(\frac{\theta}{2})} \]
Students can improve their grasp of these identities by practicing problems that require converting a trigonometric function of a double angle into single angle expressions. Doing so facilitates the analysis and graphing of trigonometric functions, like in the given exercise where the sine function is expressed through a tangent function using a double angle identity.

Practical Applications of Double Angle Identities

• Solving trigonometric equations
• Simplifying complex trigonometric expressions
• Performing integrations in calculus
• Analyzing vibrations and waves in physics

Sine Function Graph

The sine function graph is a visual representation of \(y = \sin(x)\), which is a fundamental concept in trigonometry. This graph depicts the relationship between the angle \(x\) and the sine of that angle. In a unit circle, the sine of an angle is the y-coordinate of the point corresponding to the angle on the circumference of the circle.
When graphed, the sine function produces a wave-like pattern called a sinusoid. Noticeable characteristics of this pattern are:

• It oscillates between 1 and -1.
• It has a period of \(2\pi\), meaning the wave repeats every \(2\pi\) radians.
• The wave crosses the x-axis at \(0\), \(\pi\), and \(2\pi\) within one period.
• The maxima and minima occur at quarter periods (\(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) respectively).

Key Points In Graphing a Sine Function

• Determine the amplitude, which is the height from the centerline to the peak.
• Identify the period and frequency of the oscillation.
• Locate the phase shift, which tells us if the graph is moved left or right on the axis.
• Find any vertical shifts that move the entire graph up or down.

In the case of the exercise, the graph of \(y = \sin(x)\) would reveal these characteristics clearly, making the sine function a useful tool for interpreting and predicting cyclical phenomena.

Trigonometric Graph Analysis

Trigonometric graph analysis involves studying the attributes of the graphs of trigonometric functions to understand their behavior. In the given exercise, the graph produced by the equation \(y = \frac{2 \tan \frac{x}{2}}{1+\tan^2 \frac{x}{2}}\) is to be analyzed. To do this effectively:

• Recognize the form of the trigonometric function involved.
• Identify the amplitude, period, and phase shifts if any.
• Observe any symmetry, such as even or odd functions.
• Note the intercepts and asymptotes that impact the graph's layout.

In this case, after graphing the equation, the analysis shows that the function behaves like a sine wave, which verifies that the given expression can indeed be described using the sine function equation \(y = \sin(x)\).
The verification of this equivalence demonstrates a deep understanding of trigonometric identities and enhances students' abilities to transition between different forms of trigonometric expressions. It is also evident in this exercise that understanding the behaviors of the trigonometric graphs assists not only in the analysis but also in conveying complex trigonometric relationships in simpler forms, which is an essential skill in various fields, including engineering, physics, and even finance.