Problem 6 - Algebra 2 and Trigonometry

Question

What is the period of the cosine function?

Step-by-Step Solution

Answer

The period of the cosine function is \(2\pi\).

Step 1: Understanding the Cosine Function

The cosine function is a periodic function, which means it repeats its values after a certain interval along the x-axis. The general form is: \( y = \,\cos(x) \), which has a specific period length.

Step 2: Identifying the Period

The cosine function, \( y = \cos(x) \), completes one full cycle as \( x \) goes from 0 to \( 2\pi \) radians. This interval \( [0, 2\pi] \) represents the period of the cosine function.

Step 3: Conclusion on the Period

Therefore, the period of the standard cosine function \( y = \cos(x) \) is \( 2\pi \) radians.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in trigonometry and are used to describe relationships in right-angled triangles. There are six primary trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
These functions are valuable in various branches of mathematics and applied sciences.

• Sine and cosine are the most commonly used. They represent ratios of sides in a right triangle: sine is the ratio of the opposite side over the hypotenuse, while cosine is the ratio of the adjacent side over the hypotenuse.


• Tangent is the ratio of the sine and cosine values. It gives the slope of an angle.

Trigonometric functions also help in modeling periodic phenomena such as waves and cycles. Each function has its own graph, period, and amplitude properties.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. The cosine function is a great example of this, having a specific interval over which it repeats.
In mathematical terms, if a function \( f(x) \) is periodic, there exists a non-zero constant, \( T \), such that:

• For all values of \( x \) in the domain of \( f \), \( f(x + T) = f(x) \).


• The smallest positive value of \( T \) is called the period of the function.

The cosine function, \( y = \cos(x) \), repeats every \( 2\pi \) radians. This means, after every \( 2\pi \) interval along the x-axis, the values will repeat in the same order. Periodic functions are essential for modeling and understanding recurring phenomena, such as sound waves, light waves, and seasonal temperature variations.

Radian Measure

Radian measure is an alternative to degrees for measuring angles. It is essential for understanding trigonometric functions, especially when working in a mathematical context.

• Radians relate directly to the arc length of a circle. For a circle of radius 1 (a unit circle), the angle in radians is equal to the length of the arc.
• The whole circle equates to an angle of \( 2\pi \) radians. This offers a seamless connection to the period of the cosine function, which also completes a cycle in \( 2\pi \) radians.

Radians are crucial in calculus and advanced math because they simplify the interpretation and calculation of derivatives and integrals of trigonometric functions. Switching from degree to radian measure can intimidate at first, but it becomes intuitive once you recognize its broader applications in mathematics and science.