Problem 27 - Algebra 2 and Trigonometry

Question

In \(24-32,\) find the exact value of each expression. $$ \cos (\arccos (-1)) $$

Step-by-Step Solution

Answer

The value is -1.

Step 1: Understanding the Expression

The given expression is \( \cos (\arccos (-1)) \). We need to understand what each part of the expression means. The \( \arccos \) function (inverse cosine) will give us an angle whose cosine is the input value. The input here is -1, so \( \arccos(-1) \) means finding the angle whose cosine is -1.

Step 2: Calculating \( \arccos(-1) \)

The angle whose cosine is -1 is \( \pi \) radians (or \( 180^\circ \)). This is because the cosine of \( \pi \) is -1.

Step 3: Simplifying the Expression

Now that we know \( \arccos(-1) = \pi \), we can substitute it back into the original expression. Hence, \( \cos (\arccos(-1)) = \cos(\pi) \).

Step 4: Calculating \( \cos(\pi) \)

Finally, we calculate \( \cos(\pi) \). The cosine of \( \pi \) radians is -1. Thus, the exact value of \( \cos (\arccos(-1)) \) is -1.

Key Concepts

Cosine Function

The cosine function is one of the primary functions in trigonometry, represented as \( \cos \). It describes the relationship between the angle and the length of the adjacent side over the hypotenuse in a right-angled triangle. The cosine function is periodic, meaning it repeats values in a predictable pattern. It is typically defined on the interval from 0 to \( 2\pi \) radians or 0 to 360 degrees for a standard unit circle.
Cosine has a range of values from -1 to 1. It's crucial for solving trigonometric equations and is widely used in various applications like physics for modeling waves.

• For angle \( 0 \), \( \cos(0) = 1 \)
• At \(\pi\), \( \cos(\pi) = -1 \)
• At \(\pi/2\) and \(3\pi/2\), \( \cos \) reaches 0

Understanding the basic nature of the cosine function helps immensely in dealing with trigonometric identities and inverse functions.

Angle Measurement

Angles can be measured in several units, but the most common are degrees and radians. Degrees are often used in everyday contexts, while radians are preferred in higher mathematics, especially calculus and trigonometry.
One complete revolution around a circle is \( 360^{\circ} \) or \( 2\pi \) radians, making these two units interchangeable with appropriate conversion.

• \(90^{\circ} = \pi/2\) radians
• \(180^{\circ} = \pi\) radians
• \(270^{\circ} = 3\pi/2\) radians

When working with trigonometric functions, it's important to know how to switch units. For instance, the calculation of \( \arccos(-1) \) results in an angle of \(360^{\circ}\) in degrees and \(\pi\) radians.

Radians

Radians are a unit of measure for angles that use the radius of the circle to determine the scale. By defining an angle using radians, we're saying how many radii fit along an arc that subtends the angle. One radian equals the angle made when the arc length is equal to the radius of the circle.
This measurement is more natural in calculus-based mathematics, offering smoother conversions, especially when dealing with trigonometric identities and inverse trigonometric functions.

Some important values include:

• \(\pi\) radians is equivalent to a half-turn around a circle.
• An angle of \(\pi/2\) radians (90 degrees) means the arc's length equals half the circumference of a unit circle.

When dealing with trigonometric functions, knowing the radian measures of common angles is invaluable.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the angle variables. These identities are fundamental in simplifying trigonometric expressions and solving equations. They relate the basic functions (sine, cosine, and tangent) in various ways.

• The Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal identities, linking each of the trigonometric functions to its reciprocal function.
• Cofunction identity: \( \cos(\theta) = \sin(\pi/2 - \theta) \)

Understanding these identities allows students to transform and manipulate trigonometric expressions with ease. In particular, knowing the identity for inverse functions helps when calculating values like \( \cos(\arccos(-1)) \), simplifying the task to recognizing that the expression resolves to its initial input, which is -1.