Problem 29 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{5\pi}{6} \).

Step 1: Understand the Problem

We are tasked with finding the exact value of \( \cos^{-1}\left(\cos \frac{5 \pi}{6}\right) \). The expression involves the inverse cosine (arc cosine) function, which returns an angle whose cosine is the given value.

Step 2: Recall the Range of Cosine Inverse Function

The range of the \( \cos^{-1} x \) function is from \( 0 \) to \( \pi \) radians. Because \( \frac{5\pi}{6} \) is within this range, if \( \cos \theta = \cos \frac{5\pi}{6} \), then \( \cos^{-1}(\cos x) = x \) when \( x \) is in the range \( [0, \pi] \).

Step 3: Calculate Cosine Value

Compute \( \cos \frac{5\pi}{6} \). The angle \( \frac{5\pi}{6} \) is in the second quadrant where cosine is negative. We know \( \cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} \). Since \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), it follows that \( \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \).

Step 4: Apply the Inverse Cosine Function

Now, evaluate \( \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) \). The angle whose cosine is \(-\frac{\sqrt{3}}{2}\) in the range \(0\) to \(\pi\) is \(\frac{5\pi}{6}\).

Step 5: Reach the Conclusion

Since the input angle \( \frac{5\pi}{6} \) is within the range \( [0, \pi] \), the identity \( \cos^{-1}(\cos x) = x \) applies directly, confirming that \( \cos^{-1}\left(\cos \frac{5\pi}{6}\right) = \frac{5\pi}{6} \). Thus, the expression is defined and its exact value is \( \frac{5\pi}{6} \).

Key Concepts

Trigonometric Identities

Trigonometric identities simplify the computation of trigonometric functions and their inverses. When facing an expression like \(\cos^{-1}(\cos x)\), a handy identity comes into play: if \(x\) is within the range of \([0, \pi]\), then \(\cos^{-1}(\cos x) = x\). This property is due to the nature of the cosine function and its inverse.
Here are some key trigonometric identities that are useful:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \).
• Reciprocal Identity: \( \cos\theta = \frac{1}{\sec\theta} \).
• Odd-Even Identities: \( \cos(-\theta) = \cos\theta \) and \( \sin(-\theta) = -\sin\theta \).

Understanding these identities not only helps in calculations but also deepens comprehension of trigonometric functions and their interrelationships. By mastering these, it becomes easier to predict and evaluate problems where trigonometric inverses are used, just like in this exercise.

Quadrants in Trigonometry

Trigonometric functions change signs across different quadrants, and recognizing the quadrant of an angle helps in determining these signs. A quadrant is a section of the coordinate plane, divided by the x and y axes into four parts.
The position of an angle in these quadrants affects the value of trigonometric functions:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

For example, \(\frac{5\pi}{6}\) is in the second quadrant. Because of this, \(\cos\frac{5\pi}{6}\) is negative and equivalent to \(-\cos\frac{\pi}{6}\). This knowledge is crucial when solving problems involving trigonometric identities and inverses. Recognizing the quadrant can simplify finding exact values, as in the given exercise.

Range of Trigonometric Functions

The range of a trigonometric function is the set of all possible output values. For inverse trigonometric functions, such as \(\cos^{-1} x\), the range is particularly important as it dictates which angle corresponds to a given cosine value.

• Inverse Cosine Function: The range is \([0, \pi]\). This means any angle that results from \(\cos^{-1}\) will lie within this interval.
• Why is the range restricted? It's because the inverse cosine must be a function, meaning for each input value, there's exactly one output value.

So, if the cosine of an angle is within the range, the \(\cos^{-1}\) function can return that angle. This property was applied in the original exercise to conclude \(\cos^{-1}(\cos \frac{5\pi}{6})\) equals \(\frac{5\pi}{6}\), since this angle is within the range \([0, \pi]\). Such understanding is essential for solving trigonometric problems efficiently.