Problem 94 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \cos 0 $$

Step-by-Step Solution

Answer

The exact value of \(\cos 0\) is 1.

Step 1: Understanding the Problem

We need to find the exact value of the cosine function when the angle is 0 degrees (or 0 radians). Cosine is one of the primary trigonometric functions which can be found on the unit circle.

Step 2: Identify the Position on the Unit Circle

On the unit circle, any angle is represented as a point (x, y), where x corresponds to the cosine of the angle and y corresponds to the sine of the angle. For angle 0, this point is (1, 0).

Step 3: Read the Cosine Value from the Point

From the point (1, 0), we see that the x-coordinate, which is 1, represents the cosine value of the angle. Hence, \[\cos 0 = 1.\]

Step 4: Verify with Trigonometric Identity

Using the identity \(\cos^2\theta + \sin^2\theta = 1\), substitute \(\theta = 0\). We have \(\cos^2 0 + \sin^2 0 = 1\). With \(\sin 0 = 0\), this simplifies to \(\cos^2 0 = 1\), confirming \(\cos 0 = 1\).

Key Concepts

Unit Circle

The unit circle is an essential tool in trigonometry, designed to visualize the relationships between angles and trigonometric functions. It is a perfect circle centered at the origin of a coordinate plane with a radius of one unit. This circle allows us to easily find the values of trigonometric functions for different angles.Every point on the unit circle is a pair \(x, y\), where \(x\) denotes the cosine of the angle and \(y\) represents the sine of the angle. This idea helps remember that for any angle \(\theta\), the coordinates of the point are (\(\cos\theta, \sin\theta \)). When considering angles like 0, 90, 180, or 270 degrees, the unit circle becomes particularly useful.

• At 0 degrees (or 0 radians), the point is (1, 0) on the circle.
• At 90 degrees (or \(\frac{\pi}{2}\) radians), the point is (0, 1).
• At 180 degrees (or \(\pi\) radians), the point is (-1, 0).
• At 270 degrees (or \(-\frac{\pi}{2}\) radians), the point is (0, -1).

These key reference points make it simple to remember the values of trigonometric functions at these angles, helping to solve problems like finding \(\cos 0\).

Cosine Function

Within trigonometry, the cosine function measures the adjacent side over the hypotenuse in a right triangle, but on the unit circle, it is the x-coordinate of a point at a given angle.The cosine function, \( \cos\theta\), is periodic with a period of \((2\pi)\), meaning it repeats its values every 360 degrees. This makes it a vital function for modeling periodic phenomena like waves or circular motion.For basic angles like 0 degrees (0 radians), the cosine function serves as a direct way to read off the value from the unit circle:

• At 0 degrees, the point on the unit circle is (1, 0), meaning \(\cos 0 = 1\).
• At 90 degrees, the point is (0, 1), so \(\cos 90 = 0\).
• At 180 degrees, the point becomes (-1, 0), implying \(\cos 180 = -1\).
• At 270 degrees, the point is (0, -1), which gives \(\cos 270 = 0\).

These values are essential in understanding the behavior of cosine as angles increase or decrease, confirming that \(\cos 0\) is indeed 1.

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying expressions and solving equations involving trigonometric functions. One of the most fundamental of these is the Pythagorean identity:\[\cos^2\theta + \sin^2\theta = 1\]This identity arises from the Pythagorean theorem and holds for any angle \(\theta\). It indicates that the sum of the squares of cosine and sine of the same angle is always one.For the specific case of \(\theta = 0\), the identity can be confirmed as follows:

• Substitute \(\theta = 0\) into the identity: \(\cos^2 0 + \sin^2 0 = 1\).
• Since \(\sin 0 = 0\), the equation simplifies to \(\cos^2 0 = 1\).
• This aligns with \(\cos 0 = 1\), as squaring either 1 or -1 gives 1, thus confirming the value determined using the unit circle.

The identity not only confirms our unit circle findings but also acts as a foundational tool for further understanding and manipulating trigonometric functions.