Problem 29 - Algebra and Trigonometry

Question

\(23-44=\) 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 cosine inverse function

Recall that the function \( ext{cos}^{-1}(x)\), also known as arccosine, returns the angle whose cosine is \(x\). The range of \( ext{cos}^{-1}\) is \([0, \pi]\), meaning it will give an angle between \(0\) and \(\pi\).

Step 2: Consider the input angle

The angle given in the expression is \(\frac{5\pi}{6}\). This angle is situated in the second quadrant, where cosine values are negative. The cosine of an angle in the second quadrant between \(\pi/2\) and \(\pi\) is negative.

Step 3: Calculate the cosine of the angle

Compute \(\cos\left(\frac{5\pi}{6}\right)\). Knowing that \(\cos(\pi - \theta) = -\cos(\theta)\), where \(\theta = \pi/6\), we have \(\cos(\pi - \pi/6) = -\cos(\pi/6)\). Since \(\cos(\pi/6) = \sqrt{3}/2\), it follows that \(\cos(5\pi/6) = -\sqrt{3}/2\).

Step 4: Find the arccosine value

We are asked to evaluate \(\text{cos}^{-1}\left(\cos\frac{5\pi}{6}\right)\). Since \(\frac{5\pi}{6}\) lies within the range of \([0, \pi]\), \(\text{cos}^{-1}\left(\cos\frac{5\pi}{6}\right)\) is simply \(\frac{5\pi}{6}\).

Key Concepts

Understanding Arccosine

Arccosine, often denoted as \( \cos^{-1}(x) \), is an inverse trigonometric function. It helps us find the angle whose cosine value is \( x \). For example, if you know that the cosine of an angle is 0.5, the arccosine function can tell you that the angle is \( \frac{\pi}{3} \).
The range of the arccosine function is from \( 0 \) to \( \pi \). This means the function will only return angles within this interval. Why is this important? It ensures there is only one unique answer when you want to find an angle from its cosine.

• The input to arccosine is the result from a cosine calculation and must be between -1 and 1.
• The output is the angle in radians, lying within the interval \( [0, \pi] \).

Understanding this range helps avoid errors when working with angles on problems involving inverse trigonometric functions.

The Cosine Function Basics

The cosine function, denoted as \( \cos \), is fundamental in trigonometry. It relates the angle \( \theta \) in a right triangle to the ratio of the adjacent side over the hypotenuse.
When you plot the cosine function, it creates a wave-like graph that repeats every \( 2\pi \) radians. Its values range from -1 to 1, making it very predictable.

• Cosine of \( 0 \) degrees (or \( 0 \) radians) is 1.
• Cosine of \( 90 \) degrees (or \( \pi/2 \) radians) is 0.
• Cosine of \( 180 \) degrees (or \( \pi \) radians) is -1.

These values reflect how the cosine function progresses through the quadrants on the unit circle, providing crucial information for solving trigonometric problems.

Exploring the Unit Circle

The unit circle is a circle with radius 1, centered at the origin of a coordinate plane. It's a powerful tool for understanding trigonometric functions, including sine, cosine, and their inverses.
Points on the unit circle can be represented as \( (\cos(\theta), \sin(\theta)) \). The angle \( \theta \) is measured from the positive x-axis. The unit circle helps us visualize and define the sine and cosine functions, making it easier to grasp concepts like periodicity and amplitude.

• The x-coordinate of any point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.
• Complete one full rotation around the circle, and you'll have traveled \( 2\pi \) radians.

This makes the unit circle an essential reference for solving equations like \( \cos^{-1}(\cos(x)) \), where examining angle positions can indicate direct solutions without ambiguity.

Angle Measurement in Radians and Degrees

Angles can be measured in two main units: degrees and radians. Each has its place in mathematics, but radians are most commonly used in higher-level math and calculus.

• In a full circle, there are 360 degrees.
• Alternatively, there are \( 2\pi \) radians in the same circle.
• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \).
• Conversely, to convert from radians to degrees, multiply by \( \frac{180}{\pi} \).

Radians often provide a more natural and direct measurement for describing angles in trigonometry and calculus, especially when dealing with periodic functions and derivatives. Understanding the conversion between these units can simplify solving problems that may initially appear complex, such as finding the arccosine of a particular value.