Trigonometric Function Analysis \[ y = 3 \sin(2x + 180^\circ) \]

Trigonometric Function Analysis: y = 3 sin(2x + 180°)

Theory

Trigonometric functions are widely used to describe repeating patterns such as waves and oscillations. The sine function, in particular, can be transformed to change its height, speed, and horizontal position. One important transformation is the phase shift, which moves the graph left or right along the x-axis.

Concept Explanation

In this lesson, we analyze the trigonometric function:

\[ y = 3 \sin(2x + 180^\circ) \]

This function produces a wave that oscillates between positive three and negative three. Compared to the basic sine function, the wave is taller, faster, and shifted horizontally due to the added constant inside the sine expression.

Because of the phase shift, key points on the graph occur at different angles. For example:

• At \(-90^\circ\), the value is \(0\)
• At \(-45^\circ\), the value reaches \(3\)
• At \(0^\circ\), the value returns to \(0\)

Steps or Formula

The general form of a sine function with a phase shift is:

\[ y = a \sin(kx + c) \]

From this form, we can determine the main characteristics:

• Amplitude: \(|a|\)
• Period: \(\dfrac{360^\circ}{k}\)
• Phase shift: \(-\dfrac{c}{k}\)
• Maximum value: \(|a|\)
• Minimum value: \(-|a|\)

For the function \(y = 3 \sin(2x + 180^\circ)\), we identify:

• \(a = 3\)
• \(k = 2\)
• \(c = 180^\circ\)

Example Problem

Find the amplitude, period, and phase shift of the function:

\[ y = 3 \sin(2x + 180^\circ) \]

Solution:

1. Amplitude \(= |3| = 3\)
2. Period \(= \dfrac{360^\circ}{2} = 180^\circ\)
3. Phase shift \(= -\dfrac{180^\circ}{2} = -90^\circ\)

This means the graph is shifted \(90^\circ\) to the left compared to the standard function \(y = 3 \sin 2x\).

Final Answer

The function \(y = 3 \sin(2x + 180^\circ)\) has an amplitude of 3, a period of \(180^\circ\), and a phase shift of \(-90^\circ\). It shares the same amplitude and period as \(y = 3 \sin 2x\), but differs in horizontal position. Phase-shifted sine functions are useful for modeling real-world situations such as delayed sound waves, electrical signals, and synchronized oscillations.