Problem 24 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\cos \left(\cos ^{-1} \frac{2}{3}\right)\)

Step-by-Step Solution

Answer

The value is \(\frac{2}{3}\).

Step 1: Understand the problem

The problem requires us to evaluate the expression \( \cos \left( \cos^{-1} \frac{2}{3} \right) \). This involves understanding that \( \cos^{-1} \) is the inverse cosine function.

Step 2: Recall the Inverse Cosine Identity

The identity \( \cos \left( \cos^{-1} x \right) = x \) holds for \( x \) values in the interval \([-1, 1]\). This is because the inverse cosine function, \(\cos^{-1} x\), returns the angle whose cosine is \(x\).

Step 3: Apply the Identity

Given \( \cos \left( \cos^{-1} \frac{2}{3} \right) \), we can directly apply the identity from the previous step. So, \( \cos \left( \cos^{-1} \frac{2}{3} \right) = \frac{2}{3} \).

Step 4: Verify the Solution

Check whether \( \frac{2}{3} \) lies within the interval \([-1, 1]\). Since it is within the interval, the use of the inverse cosine identity is valid, confirming our solution is correct.

Key Concepts

Cosine Function

The cosine function, often abbreviated as \( \cos \), is one of the basic trigonometric functions. It is primarily used to relate the angles of a triangle to the lengths of its sides. The function is defined for all angles,

• In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• The cosine function is periodic, meaning it repeats its values in regular intervals. Its fundamental period is \( 2\pi \).
• The cosine function is even, which implies that \( \cos(-x) = \cos(x) \) for any angle \( x \).
• The range of the cosine function is from -1 to 1, inclusive.
• It's often depicted as a wave that starts from 1 (at 0 radians) and goes down to -1 before ascending back to 1.

If you're looking at a unit circle, which is a circle with a radius of one, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle.Understanding the behaviors and properties of the cosine function is key in solving many mathematical problems, especially those involving periodic phenomena.

Inverse Cosine Identity

Inverse functions essentially reverse the effects of the original function. The inverse cosine function, denoted as \( \cos^{-1} \) or \( \text{arccos} \), helps determine an angle when you know its cosine value.

• The identity \( \cos(\cos^{-1}(x)) = x \) is pivotal for resolving expressions involving inverse indicators.
• This identity holds true when \( x \) is within the interval \([-1,1]\), because that's where the cosine function operates fully (i.e., covers all possible output values).
• The range of \( \cos^{-1} \) is \([0, \pi]\). This means it outputs angles in radians between 0 and \( \pi \), inclusive.

Consider using this identity to evaluate expressions without calculating angles directly. In the exercise, applying \( \cos(\cos^{-1}(\frac{2}{3})) = \frac{2}{3} \) demonstrates the precise application of this identity.The inverse cosine identity simplifies potentially complex evaluations by directly yielding the cosine value used in its inverse function, streamlining problem-solving in many trigonometric scenarios.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true, regardless of the angle input given to the functions. They serve as essential tools in simplifying and solving trigonometric equations. Common identities include:

• Pythagorean Identities: These are based on the Pythagorean theorem, such as \( \sin^2(x) + \cos^2(x) = 1 \).
• Angle Sum and Difference Identities: Useful for finding the trigonometric functions of sums or differences of angles. For example, \( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \).
• Double Angle Identities: These involve expressions like \( \cos(2x) = \cos^2(x) - \sin^2(x) \).
• Reciprocal Identities: Relating functions like sine and cosecant, \( \sin(x) = \frac{1}{\csc(x)} \).
• Inverse Function Identities: Such as the inverse cosine identity discussed earlier.

Trigonometric identities allow us to express complex trigonometric expressions in simplified forms. They can help solve equations, prove other mathematical theorems, or even derive further identities. Recognizing when and how to use these identities is crucial for mastering trigonometry. Always verify results by checking if they adhere to known identities or boundaries of trigonometric functions.