Problem 6 - Algebra 2

Question

Find each value. Write degree measures in radians. Round to the nearest hundredth. \(\cos \left(\cos ^{-1} \frac{2}{9}\right)\)

Step-by-Step Solution

Answer

The value is \(\frac{2}{9}\).

Step 1: Understand the Expression

The expression is \(\cos(\cos^{-1} \frac{2}{9})\). The \(\cos^{-1}\) function, or arccosine, returns an angle whose cosine is the given value, \(\frac{2}{9}\).

Step 2: Apply the Concept of Inverse Functions

Since \(\cos^{-1}\) and \(\cos\) are inverse functions, they essentially cancel each other out. This means \(\cos(\cos^{-1}(x)) = x\) as long as \(-1 \leq x \leq 1\).

Step 3: Simplify the Expression with Given Values

Using the property from Step 2, simplify the expression: \(\cos(\cos^{-1}(\frac{2}{9})) = \frac{2}{9}\).

Step 4: Verify the Range of Cosine

Verify that \(\frac{2}{9}\) is within the range of cosine values, i.e., \(-1 \leq \frac{2}{9} \leq 1\), which it is. This ensures the operation is valid.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the effect of the standard trigonometric functions. They are crucial in finding angles when the sides of a triangle are known. When we use the cosine inverse, often written as \(\cos^{-1}\), we find an angle whose cosine value matches the given input. The notation \(\cos^{-1}(x)\) seeks an angle \(\theta\) such that \(\cos(\theta) = x\). This is particularly useful when dealing with scenarios where we need to determine angles rather than side lengths.

• Basic Properties: For \(\cos^{-1}(x)\), the result is always an angle in the range \([0, \pi]\).
• Usage Note: Always ensure the value \(x\) given to \(\cos^{-1}\) satisfies \(-1 \leq x \leq 1\) because cosine values are restricted to this interval.
• Cancellation Property: When \(\cos\) and \(\cos^{-1}\) are combined, they can cancel each other out if \(x\) is within the valid range, simplifying expressions like \(\cos(\cos^{-1}(x)) = x\).

Using inverse trigonometric functions greatly enhances our ability to resolve complex problems involving angles and provides a straightforward means to directly compute angle measurements from known ratios.

Cosine Function

The cosine function is part of the fundamental trio of trigonometric functions, often abbreviated as "cos". It emerges from right-angled triangles as the ratio of the adjacent side to the hypotenuse. In terms of its unit circle definition, it corresponds to the \(x\)-coordinate of a point on the circle.

• Periodicity: Cosine is a periodic function with a period of \(2\pi\), meaning that its value repeats every \(2\pi\) radians.
• Range: Cosine values always lie between \(-1\) and \(1\). These values are essential when dealing with angles in various quadrants of the unit circle.
• Properties: It is an even function, so \(\cos(-\theta) = \cos(\theta)\).

The cosine function plays a crucial role in various applications ranging from geometry to physics, especially in wave equations and harmonic motion. Understanding its behavior and properties allow us to solve myriad mathematical problems efficiently.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, radians consider the arc length corresponding to the angles, providing a natural and mathematics-friendly way to express angles.

• Definition: A full rotation around a circle is \(2\pi\) radians, contrasting with the 360 degrees system.
• Conversion Relation: The conversion factor between degrees and radians is \(\pi/180\), facilitating easy transition between units.
• Why Use Radians? They simplify mathematical expressions in calculus, especially in derivative and integral formulas for trigonometric functions.

Using radians enhances the coherence of mathematical operations and ensures that formulae involving trigonometric functions remain consistent and easy to handle. Radians form the backbone of advanced math and are integral to effective problem-solving in various scientific disciplines.