Problem 256 - Precalculus

Question

For the following exercises, determine whether the equation is true or false. $$ \arccos \left(\cos \left(\frac{5 \pi}{6}\right)\right)=\frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The equation is true.

Step 1: Understand the Equation

The given equation is \( \arccos \left(\cos \left(\frac{5 \pi}{6}\right)\right)=\frac{5 \pi}{6} \). Our task is to check if this equation is true or false.

Step 2: Determine the Value of \(\cos\left(\frac{5\pi}{6}\right)\)

The cosine function for angles in radians is based on the unit circle. For the angle \( \frac{5\pi}{6} \), which is in the second quadrant where cosine values are negative, we have \( \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2} \).

Step 3: Evaluate \(\arccos\) of the Result

The function \( \arccos(x) \) gives the angle in the range \([0, \pi]\) whose cosine is \(x\). Since \( \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2} \), we apply \( \arccos \) to this value to find the angle. Therefore, \( \arccos \left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6} \), because \(\frac{5\pi}{6}\) is the angle within that range where its cosine is \(-\frac{\sqrt{3}}{2}\).

Step 4: Check the Truth of the Equation

Since \( \arccos \left(\cos \left(\frac{5 \pi}{6}\right)\right) = \frac{5 \pi}{6} \), the equation remains true when evaluated. Therefore, the equation is true.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the trigonometric values are known. They essentially reverse the process of the standard trigonometric functions. The inverse cosine function, denoted as \( \arccos(x) \), gives the angle whose cosine value is \( x \).

• For the cosine function, the range of its output values is from -1 to 1.
• \( \arccos(x) \) maps these values back to an angle in radians within the principal range of [0, \( \pi \)], covering the first and second quadrants of the unit circle.

Understanding this reversal is crucial in problems where we find the angle corresponding to a given cosine value. For example, in our exercise, we used \( \arccos \) to determine the angle that has a cosine of \( -\frac{\sqrt{3}}{2} \). This operation allowed us to establish the angle as \( \frac{5\pi}{6} \), confirming the truth of the equation.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle is used to define trigonometric functions for all real numbers. Angles on the unit circle can be measured in radians or degrees, but radians are often used for their natural mathematical properties.

• The x-coordinate of a point on the unit circle corresponds to the cosine of the angle formed with the positive x-axis.
• The y-coordinate corresponds to the sine of that angle.
• The unit circle allows trigonometric functions to be applied across all four quadrants:
  • Quadrant I (0 to \( \frac{\pi}{2} \)): both sine and cosine are positive.
  • Quadrant II (\( \frac{\pi}{2} \) to \( \pi \)): sine is positive, cosine is negative.

In our exercise, \( \frac{5\pi}{6} \) is an angle in the second quadrant, where cosine values are negative, confirming the result of \( \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}. \)

Cosine Function

The cosine function is one of the primary functions in trigonometry. This function relates the angle in the unit circle to the x-coordinate of the corresponding point. The cosine of an angle \( \theta \) is defined as the x-value of the point where the angle intersects the unit circle.

• In mathematical terms, the cosine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
• The cosine function is even, which means \( \cos(-\theta) = \cos(\theta) \).

The cosine function is critical for solving problems related to wave patterns, circular motion, and real-world phenomena. In our exercise, the cosine function was applied to \( \frac{5\pi}{6} \), resulting in a value of \( -\frac{\sqrt{3}}{2} \), which falls in line with the expected result for an angle situated in the second quadrant of the unit circle.