Question

Prove each identity. $$\cos (2 \pi-x)=\cos x$$

Short answer

\( \cos (2 \pi - x) = \cos x \) is proven by applying the cosine angle subtraction identity.

Step-by-step solution

Identify the cosine angle subtraction identity

We will use the cosine angle subtraction identity, which states that \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \). Here, \( \alpha = 2 \pi \) and \( \beta = x \).

Apply the angle subtraction identity

Using the identity on \( \cos (2 \pi - x) \), we get \( \cos (2 \pi - x) = \cos 2 \pi \cos x + \sin 2 \pi \sin x \).

Evaluate the cosine and sine of \(2\pi\)

\( \cos 2 \pi \) is equal to 1 because it represents the cosine of one full rotation around the unit circle, and \( \sin 2 \pi \) is equal to 0 because it represents the sine at the same point.

Simplify the expression

Substituting the values, we get \( \cos 2 \pi \cos x + \sin 2 \pi \sin x = 1 \cdot \cos x + 0 \cdot \sin x = \cos x \).

Conclude the proof

Since we have shown that \( \cos (2 \pi - x) = \cos x \), the identity is proven.

Key concepts

Cosine Angle Subtraction Identity

Understanding the cosine angle subtraction identity is fundamental for simplifying complex trigonometric expressions. This identity states that for any two angles, \( \alpha \) and \( \beta \) the equation \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \) holds true.

It's a tool that allows us to deconstruct the cosine of a difference between two angles into a form involving only the cosines and sines of the individual angles. This identity is particularly useful for solving trigonometric equations, proving other identities, and even in calculations involving vectors and complex numbers. To properly use it, we must recognize the structure \( \cos(\alpha - \beta) \) and apply the identity correctly. Additional understanding of the unit circle and the values of cosine and sine for key angles can greatly help in simplifying the results.

For instance, in the exercise, when asked to prove \( \cos (2 \pi-x) = \cos x \) we used this identity by setting \( \alpha = 2 \pi \) and \( \beta = x \) to display how subtracting angle \( x \) from a full rotation (\( 2 \pi \) radians) essentially brings us back to our starting point, hence why the cosine value remains unchanged. This identity is a powerful concept that demonstrates the cyclical nature of trigonometric functions and their deep connection with the geometry of circles.

Unit Circle

The unit circle is a cornerstone of trigonometry. It's a circle with a radius of one unit, centered at the origin of a coordinate plane. The unit circle enables us to define the trigonometric functions for all real numbers and gives a geometric interpretation of these functions in terms of angles.

Each point on the unit circle has coordinates \( (\cos \theta, \sin \theta) \), where \( \theta \) is the angle in radians between the positive x-axis and the line segment from the origin to the point. We can use these coordinates to find the values of sine, cosine, and tangent for any angle.

Furthermore, the unit circle shows us that sine and cosine values are cyclical, repeating every \( 2\pi \) radians since this is the circumference of the unit circle. This cyclical property was applied in the textbook exercise to deduce that \( \cos 2\pi = 1 \) and \( \sin 2\pi = 0 \). Understanding the unit circle leads to a more intuitive grasp of why certain trigonometric identities work, like the one we used in the aforementioned proof.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are vital in linking angles to ratios of side lengths in right triangles, and they extend these relationships to circles and waves. These functions allow us to describe periodic phenomena, analyze sound and light waves, and solve problems in engineering and physics.

Each trigonometric function has specific properties and graphs that reveal periodic trends and symmetries. The sine function, for instance, starts at zero, increases to 1 at \( \frac{\pi}{2} \) radians (90 degrees), decreases back to zero at \( \pi \) radians (180 degrees), and so on. Cosine, on the other hand, starts at 1, decreases to zero at \( \frac{\pi}{2} \) radians, and continues this pattern in a symmetrical manner across the unit circle.

Application in Proofs

In proving identities, we often use the properties and values of these functions at significant angles, like 0, \( \frac{\pi}{2} \) , \( \pi \) , and \( 2\pi \) radians. By understanding both the functions and the unit circle, we can deduce the behavior of these functions around the circle and across its quadrants. In the exercise, recognizing that \( \sin 2\pi \) equals zero and \( \cos 2\pi \) equals one was crucial to simplifying the expression and proving the identity. Trigonometric functions are interconnected and mastering their relationships, as well as their individual behaviors, is key to advancing in the study of mathematics.