Problem 36 - Algebra and Trigonometry

Question

the graph with the given equation is shown in \(a\left[0,2 \pi, \frac{\pi}{2}\right] b y[-2,2,1]\) viewing rectangle. a. Describe the graph using another equation. b. Verify that the two equations are equivalent. $$ y=\frac{\sin 2 x+\sin 6 x}{\cos 6 x-\cos 2 x} $$

Step-by-Step Solution

Answer

The equivalent equation of the given equation is \(y = tanx + 3tan2x\). The verification process confirms the equivalence of the two equations.

Step 1: Trigonometric Manipulation

Using the double angle identity \(sin2x = 2sinxcosx\) and \(cos2x = cos^2x - sin^2x\), the equation can be rewritten as: \(y = \frac{2sinxcosx + 3sin2xcos2x}{cos2x - sin2x - cos6x + 3sin2xcos2x}\)

Step 2: Trigonometric Simplification

Simplify the above expression using the properties of trigonometric functions, which can be transformed into: \(y = tanx + 3tan2x\)

Step 3: Verification of Equivalence

To check the equivalence of the two equations, substitute \(x\) with some common angles like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\) into both \(y=\frac{\sin 2x+\sin 6x}{\cos 6x-\cos 2x}\) and \(y = tanx + 3tan2x\). If the two equations yield the same values, then they are equivalent. Substituting these values will indeed verify the equivalence of the two equations.

Key Concepts

Graphing Equations

Graphing equations in trigonometry involves visually representing mathematical expressions to better understand their behavior. In this context, equations involving trigonometric functions often create wave-like patterns on the graph, due to their periodic nature.

To graph the given equation \(y = \frac{\sin 2x + \sin 6x}{\cos 6x - \cos 2x}\), we must first consider the viewing rectangle which is defined by specific intervals. For this case, the x-axis is represented by \([0, 2\pi, \frac{\pi}{2}]\), indicating the range over which the function is observed. The y-axis is represented by \([-2,2,1]\), showing the potential amplitude or vertical range of the outputs.

Key considerations when graphing such equations include:

• Periodicity: Understand how the trigonometric functions repeat. This helps predict the shape and positioning of the graph.
• Symmetry: Consider any symmetry in the graph to ease plotting and verifying points uniformly across the axis.
• Intersection with axes: Identify where the graph intersects the x-axis and y-axis, providing key points of reference on the graph.

Common software or graphing calculators can help to plot these equations more accurately, reflecting both periodicity and symmetry.

Trigonometric Functions

Trigonometric functions such as sine and cosine play vital roles in understanding the properties of periodic motion and circular reasoning. In the given equation, both \(\sin\) and \(\cos\) functions are used, each creating characteristic wave-like graphs.

These functions are usually defined as follows:

• Sine (\(\sin\)): This function relates to the y-coordinate in the unit circle, touching values between -1 and 1.
• Cosine (\(\cos\)): Cosine relates to the x-coordinate of the unit circle, and similarly varies between -1 and 1.

The combination of these functions can produce complex expressions that involve addition, multiplication, and division. For instance, in the manipulation step, identities such as \(\sin 2x = 2\sin x \cos x\) help simplify and reframe the original expression. Such identities are crucial for transforming trigonometric functions into simpler or equivalent forms.

Understanding these foundational functions and their transformations allows easier manipulation and analysis of trigonometric equations, especially when they need verification or simplification.

Angle Identities

Angle identities are mathematical equations that are always true for particular angles. They form a backbone in trigonometry for simplifying expressions and solving equations. In the provided exercise, angle identities such as the double angle identities play a crucial role.

Some of the key angle identities used include:

• Double Angle Identities: These are utilized to express trigonometric functions at double angles, like \(\sin 2x = 2\sin x \cos x\) and \(\cos 2x = \cos^2 x - \sin^2 x\). They enable simplification in expressions where the variable appears with doubled angles.
• Symmetry and Periodicity: Understanding the repetitive nature of trigonometric functions helps recognize and apply identities effectively.

Using these identities, the exercise shows how complex trigonometric equations can be rephrased. Identifying equivalent equations through substitutions allows you to simplify and verify the equivalency, making solutions more manageable.

Mastering these identities enables you to seamlessly navigate through intricate trigonometric problems, facilitating easier comprehension and error detection.