Problem 6 - Algebra and Trigonometry

Question

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \text { (a) } \cos ^{-1}\left(\frac{\sqrt{2}}{2}\right) \quad \text { (b) } \cos ^{-1} 1 \quad \text { (c) } \cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{4} \); (b) 0; (c) \( \frac{3\pi}{4} \).

Step 1: Identify the Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1} \), returns the angle whose cosine is the specified value. It outputs angles in the range \([0, \pi]\) radians.

Step 2: Solve for \( \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) \)

The inverse cosine of \( \frac{\sqrt{2}}{2} \) corresponds to an angle whose cosine value is \( \frac{\sqrt{2}}{2} \). In the principal range \([0, \pi]\), this angle is \( \frac{\pi}{4} \).

Step 3: Solve for \( \cos^{-1} 1 \)

The inverse cosine of 1 returns the angle whose cosine value is 1. In the principal range \([0, \pi]\), this angle is 0, since \( \cos(0) = 1 \).

Step 4: Solve for \( \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) \)

The inverse cosine of \(-\frac{\sqrt{2}}{2}\) corresponds to an angle whose cosine value is \(-\frac{\sqrt{2}}{2}\). In the range \([0, \pi]\), this angle is \( \frac{3\pi}{4} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. It is especially useful when dealing with periodic functions like waves. One of the primary functions in trigonometry is the cosine function, which helps us understand the ratio of the adjacent side to the hypotenuse in a right triangle. In this context, we use the inverse cosine function, often denoted as \( \cos^{-1} \), which allows us to find the angle when we’re given a cosine value. This function is essential in solving problems where we are required to find angles from specific cosine values, particularly within specified ranges.

Principal Range

When dealing with inverse trigonometric functions, understanding the principal range is crucial. The principal range is the set of output angles for which an inverse trigonometric function outputs its values. For the inverse cosine function, \( \cos^{-1} \), the principal range is \([0, \pi]\) radians. This range is specifically chosen because it ensures that each cosine value maps to exactly one angle, making the function a proper inverse. By limiting the range from 0 to \( \pi \), we ensure the function's outputs are consistent and well-defined. This concept is important for ensuring clear and unambiguous results when solving equations involving the inverse cosine function.

Radians

Radians are the standard unit of angular measure used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the radius of the circle. One complete circle is \( 2\pi \) radians, which equals 360 degrees. Thus, \( \pi \) radians is equivalent to 180 degrees, and \( \frac{\pi}{2} \) radians are equivalent to 90 degrees. When using trigonometric functions, radians are often preferred because they provide a more natural and direct relationship between the angle and the arc length on a circle. Understanding this measurement is vital, especially in problems involving inverse trigonometric functions, as solutions are typically given in radians within the principal range.