Problem 24 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \cos \left(\cos ^{-1}\left(-\frac{4}{5}\right)\right) $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{4}{5}\).

Step 1: Understand the Inverse Function

The expression \( \cos \left(\cos^{-1}\left(-\frac{4}{5}\right)\right) \) involves an inverse trigonometric function, \( \cos^{-1}(x) \), which returns the angle whose cosine is \( x \).

Step 2: Apply Inverse Cosine Property

By the property of inverses, \( \cos(\cos^{-1}(x)) = x \) for all \( x \) in the domain of \( \cos^{-1}(x) \), which is \([-1, 1]\).

Step 3: Calculate the Expression Value

Given \( x = -\frac{4}{5} \), and knowing \( \cos(\cos^{-1}(x)) = x \), substitute \( x \) into the equation: \( \cos(\cos^{-1}(-\frac{4}{5})) = -\frac{4}{5} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special functions in mathematics that help us find angles when we know the value of the trigonometric ratio. They are essentially the reverse processes of the trigonometric functions themselves. For instance, the inverse of the cosine function is denoted as \( \cos^{-1}(x) \), also known as arccosine. When you apply \( \cos^{-1} \) to some number \( x \), it will return the angle \( \theta \) such that \( \cos(\theta) = x \). This is very useful when you know a cosine value and want to find the corresponding angle.

Some properties of inverse trigonometric functions include the ability to understand relationships between angles and ratios without relying on complicated computations. They help in solving right triangles and in understanding periodic phenomena. The angles produced by inverse trigonometric functions typically belong to specific intervals to ensure one unique result. For example, the range of \( \cos^{-1}(x) \) is \([0, \pi] \) (or [0°, 180°] in degrees), which ensures that each input corresponds to a single angle.

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. It's commonly used to relate the angle of a right triangle to the lengths of the two sides adjacent to the angle. The cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. This is represented as \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

As a part of the unit circle, the cosine function represents the x-coordinate of a point on the circle as the angle measured from the positive x-axis increases. Its value ranges from -1 to 1. The cosine function is periodic, with a period of \( 2\pi \), meaning the function repeats every \( 2\pi \) radians.

Cosine is used extensively in various fields, from physics to engineering, because it analyzes wave patterns, oscillations, and many other phenomena in nature. For trigonometric identities, it pairs with sine to express key relationships and symmetries in mathematics.

Domain of Cosine

The domain of a function is the complete set of possible values of the independent variable. For the cosine function, however, it's essential to know its domain and how it works with its inverse. For the standard cosine function, its domain is the entire set of real numbers. This is because cosine is defined for any angle, even beyond the typical 0 to 360 degrees or 0 to \( 2\pi \) radians.

As for the inverse cosine function \( \cos^{-1}(x) \), its domain is restricted to \( x \) values between -1 and 1, inclusive. This is due to the fact that the cosine function outputs values only in this range; hence, these are the only real numbers you can provide as input to \( \cos^{-1} \).

• The domain of \( \cos(x) \) is \( (-\infty, \infty) \)
• The domain of \( \cos^{-1}(x) \) is \([-1, 1]\)

Understanding these domains is crucial for calculations since they dictate valid inputs for either function. When dealing with expressions like \( \cos(\cos^{-1}(x)) \), ensuring \( x \) is within this domain will guarantee the expression is valid, leading directly to comfortable problem-solving.