Problem 15 - Algebra 2 and Trigonometry

Question

Find the period of each function. \(y=\cos \frac{1}{2} x\)

Step-by-Step Solution

Answer

The period of \( y = \cos \frac{1}{2} x \) is \( 4\pi \).

Step 1: Identify the standard form

The standard form of a cosine function is \( y = \cos(bx) \). In this function, \( y = \cos \frac{1}{2} x \), \( b \) is \( \frac{1}{2} \).

Step 2: Determine the formula for period

The period of a cosine function \( y = \cos(bx) \) is calculated using the formula \( \frac{2\pi}{b} \).

Step 3: Substitute into the formula

Substitute \( b = \frac{1}{2} \) into the period formula: \( \frac{2\pi}{\frac{1}{2}} \). Simplify the expression to find the period.

Step 4: Simplify the expression

To simplify \( \frac{2\pi}{\frac{1}{2}} \), multiply \( 2\pi \) by the reciprocal of \( \frac{1}{2} \), which is 2. This gives \( 2\pi \times 2 = 4\pi \).

Step 5: Conclude the period

The period of the function \( y = \cos \frac{1}{2} x \) is \( 4\pi \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships within triangles, particularly right-angled triangles, and they also describe periodic phenomena like waves. The primary trigonometric functions include sine, cosine, and tangent. Each function has its own graph and specific characteristics.
These functions are periodic, meaning they repeat their values in regular intervals over time or angle, making them incredibly useful in fields such as physics, engineering, and astronomy.

• **Sine (sin)** - It relates the angle to the opposite side over the hypotenuse of a right triangle.
• **Cosine (cos)** - It connects the angle to the adjacent side over the hypotenuse.
• **Tangent (tan)** - It's the ratio of the opposite side to the adjacent side.

The cosine function, in particular, is useful because it starts at its maximum value when the angle is zero and decreases to its minimum. It oscillates smoothly between 1 and -1. This oscillation makes cosine applicable in modeling waves and vibrations. Understanding these functions aids in solving complex problems in trigonometry and beyond.

Standard Form of Cosine

The standard form of the cosine function is essential for easily identifying its characteristics, including amplitude, period, and phase shift. The typical representation of cosine is written as:
\[ y = a \cdot \cos(bx + c) + d \]
Here's what each parameter represents:

• **"a"**: This is the amplitude, which affects the vertical stretch or compression of the graph. It determines the height from the midline to the peak or trough.
• **"b"**: This modifies the period of the function. Higher values of "b" decrease the period, making the waves go more quickly.
• **"c"**: This value introduces a horizontal shift, moving the graph left or right.
• **"d"**: This shifts the graph vertically, moving it up or down.

In the exercise, the given function is \( y = \cos \left( \frac{1}{2} x \right) \), where "b" is given as \( \frac{1}{2} \). This indicates the function will have a different period compared to the usual cosine function, which typically has a period of \( 2\pi \). Understanding the standard form helps in graph plotting and predictions for the cosine function.

Period of a Function

The period of a function, especially in trigonometric terms, describes how often the function repeats its values. For periodic functions like sine and cosine, it's the length of one complete cycle of the wave before it starts repeating again. This is a crucial concept for modeling repetitive behavior such as sound waves, light, and tides.
For the cosine function, specifically, the formula to calculate the period is:

• **Period =** \( \frac{2\pi}{b} \)

Here, "b" is the coefficient of "x" within the cosine function, measured in radians. Calculating the period helps us understand how stretched or compressed the function's cycle will be on a graph.
In the exercise, the given function is \( y = \cos \left( \frac{1}{2} x \right) \). By substituting \( b = \frac{1}{2} \) into the period formula, we calculate: \[ \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi \]
Thus, the function repeats every \( 4\pi \) units along the x-axis. Visualization of the function's graph helps solidify understanding and application in various contexts.