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.

Trigonometric Function Analysis \[ y = 3 \sin 2x \]

Trigonometric Function Analysis: y = 3 sin 2x

Theory

Trigonometric functions are commonly used to describe periodic motion and wave patterns. One of the most important trigonometric functions is the sine function, which produces a smooth, repeating wave. By changing certain parameters, we can adjust the height and speed of this wave.

Concept Explanation

In this lesson, we analyze the function:

\[ y = 3 \sin 2x \]

This function is a transformed version of the basic sine function \(y = \sin x\). The graph oscillates between positive three and negative three, creating a taller and faster wave compared to the standard sine curve.

Some important points on the graph include:

• At \(0^\circ\), the value is \(0\)
• At \(45^\circ\), the value reaches \(3\)
• At \(90^\circ\), the value returns to \(0\)
• At \(135^\circ\), the value reaches \(-3\)
• At \(180^\circ\), the function completes half a cycle

Steps or Formula

The general form of a sine function is:

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

From this formula, we can determine the key characteristics:

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

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

• \(a = 3\)
• \(k = 2\)

Example Problem

Determine the amplitude and period of the function:

\[ y = 3 \sin 2x \]

Solution:

• Amplitude \(= |3| = 3\)
• Period \(= \dfrac{360^\circ}{2} = 180^\circ\)
• Maximum value \(= 3\)
• Minimum value \(= -3\)

This means the wave completes one full cycle every \(180^\circ\), which is twice as fast as the basic sine function.

Final Answer

The function \(y = 3 \sin 2x\) has an amplitude of 3 and a period of \(180^\circ\). Compared to \(y = \sin x\), it oscillates twice as fast and reaches higher maximum and minimum values. Such functions are useful for modeling real-world oscillating systems, such as mechanical vibrations and mass-spring motions.

Dunia Segitiga: Sains, Petualangan, dan Kejutan!

Triangle Adventure: Fun & Easy Math for Curious Kids!

This interactive eBook helps children understand triangles in a fun, simple, and meaningful way. Instead of memorizing formulas, kids are guided to see how triangles appear in everyday life while building strong mathematical thinking skills step by step.

Through engaging stories, young learners will explore geometry concepts alongside friendly characters who introduce ideas like Pythagorean logic and early trigonometry in a way that feels natural and exciting. Each section combines storytelling, clear explanations, and kid-friendly examples that make learning enjoyable.

Designed for elementary to early middle school students, this eBook encourages creativity, problem-solving, and curiosity through mini challenges, logical missions, and simple projects. It’s perfect for children who want to enjoy math while developing confidence and imagination.

Turn learning geometry into an adventure your child will love and remember!

📘 Get it now!