Problem 33 - Algebra 2

Question

If \(\sin 2 A=\sin 2 B,\) must \(A=B ?\) Explain.

Step-by-Step Solution

Answer

No, \(A\) does not necessarily need to be equal to \(B\). Instead, \(A\) could be \(B\), \(B + n\pi\), or \(B - n\pi\) for any integer \(n\).

Step 1: Understand Sine Function

In a unit circle, sine function gives the y-coordinate of a point on the circle corresponding to a given angle. It's important to know that sine is a periodic function which repeats itself every \(2\pi\) radians or 360 degrees. This means that \(\sin(A) = \sin(A + 2n\pi)\) where \(n\) is an integer.

Step 2: Analyze sine function of twice an angle

The problem presents us with \(\sin 2A = \sin 2B\). It's crucial to remember that this is also true: \(\sin 2A = \sin (2B+ 2n\pi)\) or \(\sin 2A = \sin (2B - 2n\pi)\), according to the property of sine function.

Step 3: Formulate the condition for \(A\) and \(B\)

So, given \(\sin 2A = \sin 2B\), \(A\) is not necessarily equal to \(B\). Instead, \(A\) can be equal to \(B\), \(B + n\pi\), or \(B - n\pi\) where \(n\) is any integer. This follows from the periodicity of the sine function and the fact that the sine value repeats every \(2\pi\) radians.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry and describes a relationship within a right-angled triangle. For any given angle, sine provides the ratio of the length of the side opposite the angle to the hypotenuse. When we use the unit circle, it shows us the y-coordinate of a point corresponding to a specific angle on the circle.

Here's why the unit circle is handy: every angle is derived from the center of the circle, with a radius of 1. This makes it easier to visualize and calculate trigonometric functions.

• At 0 radians, \ \( \sin(0) = 0 \ \).
• At \ \( \pi/2 \ \) radians (90 degrees), \ \( \sin(\pi/2) = 1 \ \).
• At \ \( \pi \ \) radians (180 degrees), \ \( \sin(\pi) = 0 \ \).
• At \ \( 3\pi/2 \ \) radians (270 degrees), \ \( \sin(3\pi/2) = -1 \ \).

Understanding this repetitive nature of the sine helps us in problems like comparing \ \( \sin 2A \ \) to \ \( \sin 2B \ \).

Exploring the Unit Circle

The unit circle is a powerful tool in trigonometry that represents all possible angles on a coordinate plane. With a radius of 1, every point on the circle matches a specific angle's cosine and sine values.

The circle is centered at the origin (0,0) and helps us visualize angles and their trigonometric results. Here's how it works:

• The angle is measured starting from the positive x-axis.
• Counterclockwise rotation is for positive angles, while clockwise is for negative ones.
• Every complete circle is \ \( 2\pi \ \) radians or 360 degrees.

This setup means the sine and cosine values repeat after every \ \( 2\pi \ \) radians, which is essential when dealing with periodic functions like sine.

Periodicity of Trigonometric Functions

One of the key features of trigonometric functions, like sine, is periodicity. This means the function repeats its values in regular intervals.

For sine, the period is \ \( 2\pi \ \) radians or 360 degrees. This property is encapsulated in the identity: \ \( \sin(\theta) = \sin(\theta + 2n\pi) \ \) where \ \( n \ \) is any integer. This implies several important things:

• If \ \( \sin 2A = \sin 2B \ \), it's not simply because \ \( A = B \ \).
• Often, \ \( A \ \) could also equal \ \( B + n\pi \ \) or \ \( B - n\pi \ \), indicating there are many possibilities beyond \ \( A = B \ \).

Understanding periodicity is crucial to grasp why multiple angles can produce the same sine value. This characteristic is used in many mathematical and real-world applications.