Problem 12 - Precalculus Mathematics for Calculus

Question

Solve the given equation. $$\cos \theta=0.32$$

Step-by-Step Solution

Answer

\( \theta \approx \\cos^{-1}(0.32) \) and \( \theta \approx 2\pi - \\cos^{-1}(0.32) \), with periodicity \( 2\pi n \).

Step 1: Understand the Problem

We need to find the angle \( \theta \) such that \( \cos \theta = 0.32 \). The cosine function gives the adjacent side over hypotenuse ratio in a right triangle or is used to find the angle in trigonometric equations.

Step 2: Use the Inverse Cosine Function

To find \( \theta \), apply the inverse cosine function (arccos) to both sides of the equation: \( \theta = \cos^{-1}(0.32) \). This will provide the principal value of \( \theta \) in radians.

Step 3: Solve for \( \theta \) Using a Calculator

Use a scientific calculator to compute \( \theta = \cos^{-1}(0.32) \). Ensure your calculator is in the correct mode (radians or degrees) as required by the problem context.

Step 4: Identify All Possible Solutions

Since cosine is positive in both the first and fourth quadrants, the general solution should include both: \( \theta = \cos^{-1}(0.32) \) and \( \theta = 2\pi - \cos^{-1}(0.32) \) when working in radians (or in terms of degrees, find \( \theta = 360^\circ - \cos^{-1}(0.32) \)).

Step 5: General Solution

For the general solution in radians: \[ \theta = \cos^{-1}(0.32) + 2\pi n \text{ and } \theta = 2\pi - \cos^{-1}(0.32) + 2\pi n \] where \( n \) is an integer representing the periodicity of the cosine function.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in finding angles from given trigonometric ratios. These functions basically reverse the operation of standard trigonometric functions. For example, when we use the inverse cosine function, denoted as \( \cos^{-1} \) or "arccos," it allows us to determine the angle \( \theta \) when the cosine value is known. In our exercise, we have \( \cos \theta = 0.32 \). By applying \( \cos^{-1}(0.32) \), we derive the principal angle whose cosine is 0.32.

It is important to remember:

• The range of \( \cos^{-1} x \) is 0 to \( \pi \) radians, or 0 to 180 degrees.
• This range represents angles typically found in the first and fourth quadrants for the cosine function, as these are where the cosine values are positive.

This makes inverse functions extremely useful in trigonometry and solving problems where the goal is to find specific angle measures from trigonometric equations. They bridge the gap between known ratios and unknown angles, helping us solve these kinds of problems.

Cosine Function

The cosine function, written as \( \cos \theta \), is one of the fundamental trigonometric functions. It describes the relationship between the angle \( \theta \) in a right triangle and the ratio of the lengths of the adjacent side to the hypotenuse. This property extends to its usage in the unit circle.On the unit circle:

• \( \cos \theta \) equals the x-coordinate of the point where the terminal side of the angle \( \theta \) intersects the circle.
• The function is positive in the first and fourth quadrants, meaning it returns positive values for angles within these ranges.

Cosine is also periodic with a period of \( 2\pi \) radians (or 360 degrees), meaning it repeats its values every full circle around the circle. This periodic nature is crucial when identifying multiple angle solutions for trigonometric equations involving the cosine function because it allows for infinite solutions when adding multiples of \( 2\pi \).

Understanding the cosine function's behavior is key to effectively solving problems like our exercise.

Angle Solutions

In trigonometric equations, finding solutions for the angle \( \theta \) involves more than just calculating one answer. Given that trigonometric functions are periodic, multiple valid solutions exist beyond the principal angle. The cosine equation \( \cos \theta = 0.32 \) provides us with a clear example of how to approach and identify these solutions.Steps to find angle solutions:

• Calculate the principal value using the inverse trigonometric function, i.e., \( \theta = \cos^{-1}(0.32) \).
• Recognize that since cosine is positive in both the first quadrant and the fourth quadrant, there are at least two immediate angle solutions: one found by the inverse cosine and another symmetrical one in the fourth quadrant.

To capture all possible solutions, leverage the periodic nature of the cosine function. This entails:

• Adding multiples of the full circle, \( 2\pi \), to each solution: \( \theta = \cos^{-1}(0.32) + 2\pi n \) and \( \theta = 2\pi - \cos^{-1}(0.32) + 2\pi n \), where \( n \) is any integer.

Understanding angle solutions means acknowledging the inherent cyclic behavior of trigonometric functions, thus ensuring comprehensive problem-solving strategies.