Problem 32 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \cos \left(\cos ^{-1} 0.57\right) $$

Step-by-Step Solution

Answer

The exact value of the given expression \(\cos (\cos^{-1}(0.57))\) is 0.57.

Step 1: Understand the inverse cosine function

The inverse cosine function, denoted as \(\cos^{-1}(x)\) or arccos(x), is a function that takes a number \(x\) between -1 and 1 (inclusive) and returns an angle, measured in radians, whose cosine is \(x\).

Step 2: Apply the inverse cosine function

In this case, \(\cos^{-1}(0.57)\) denotes the angle whose cosine is 0.57. This means that the cosine of that angle is 0.57 by definition.

Step 3: Find the value of the given expression

Now, \(\cos(\cos^{-1}(0.57))\) denotes the cosine of the angle from step 2. Since by definition, the cosine of that angle is 0.57, the value of the given expression is 0.57.

Key Concepts

Inverse Cosine

The inverse cosine, often notated as \( \cos^{-1}(x) \) or arccos(x), is a special function in trigonometry. It helps us find the angle whose cosine is a given number. The input value \( x \) must lie between -1 and 1, as those are the only possible values for cosine of any angle. This range is necessary because the cosine values do not exceed this range for angles measured in radians or degrees.

• Understanding Inverse Functions: Inverse functions essentially 'reverse' the effect of the original function. For instance, while the cosine function takes an angle and gives you the cosine value, the inverse cosine takes that cosine value and provides you back the angle.
• Common Uses: Whenever you know the cosine value but not the corresponding angle, you can use \( \cos^{-1}(x) \) to find an angle that has that particular cosine value.

Think of \( \cos^{-1} \) as a tool for solving the puzzle of "What angle has this cosine?" By using the inverse cosine, you solve this efficiently. In our exercise, we use \( \cos^{-1}(0.57) \) to find the angle where the cosine is 0.57. By the nature of the inverse function, plugging this angle back into the cosine yields the original value of 0.57.

Trigonometric Identities

Trigonometric identities are crucial tools in simplifying expressions and solving problems in mathematics. These identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the identity are defined. They help in transitioning from complex trigonometric expressions to more manageable forms.

• Basic Identities: The basic trigonometric identities include sine, cosine, and tangent, each with their reciprocal identities: cosecant, secant, and cotangent.
• Functional Identity: When you use \( \cos (\cos^{-1} x) \), you are effectively applying an identity that returns the original \( x \) if \( x \) is in the valid range of -1 to 1.

These identities, including the inverse and functional identities, can be powerful. In acute mathematics, they ensure that the transformations maintain correctness. In our exercise, using such identities confirms that \( \cos (\cos^{-1} 0.57) \) equals 0.57, preserving the function cycle integrity.

Cosine Function

The cosine function is a fundamental trigonometric function used widely in mathematics to represent the ratio of the length between the adjacent side and the hypotenuse of a right-angle triangle. It takes an angle as input, usually in radians or degrees, and returns a value between -1 and 1. This range is significant because any angle's cosine will never exceed these values.

• Cosine of Angles: Each angle has a specific cosine value, and for common angles like \(0\), \( \frac{\pi}{2} \), \(\pi \), etc., cosine has well-known values: \( \cos(0) = 1 \), \( \cos(\frac{\pi}{2}) = 0 \), \( \cos(\pi) = -1 \).
• Graph Characteristics: The cosine graph is periodic with a period of \( 2\pi \), indicating it repeats every \( 2\pi \). This is central to understanding the behavior of cosine over different intervals.

The cosine function also plays a vital role in various fields such as engineering, physics, and even in real world problem solving where waves or circular motion is involved. In the exercise, recognizing how cosine retrieves the original value of its inverse exemplifies its precision and reliability in retaining and interpreting information within its defined scope.