Problem 69 - Algebra and Trigonometry

Question

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$ \cos x=0.4 $$

Step-by-Step Solution

Answer

The solution is \( x \approx 1.16 \) radians.

Step 1: Understand the Problem

We are tasked with finding all solutions to the equation \( \cos x = 0.4 \) within the interval \([0, \pi]\). We need to use a calculator or inverse trigonometric functions to find these solutions. The answers should be correct to two decimal places.

Step 2: Determine Principal Value Using Inverse Cosine

Use the inverse cosine function to find the principal value of \( x \) where \( \cos x = 0.4 \). Compute \( x = \cos^{-1}(0.4) \). Using a calculator, we find that \( x \approx 1.15927948 \). Round this to two decimal places to get \( x \approx 1.16 \).

Step 3: Identify All Solutions in the Interval

Since \( \cos x \) is positive in the interval \([0, \pi]\), there is no secondary angle to consider within this specific interval aside from the principal angle we found. Thus the primary solution \( x = 1.16 \) is the only one within the given interval.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is defined in the context of a right-angled triangle or the unit circle. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. This is often expressed as:

• Adjacent/Hypotenuse

On the unit circle, the cosine of an angle measures the horizontal distance from the origin to the point on the circle defined by the given angle.
In terms of a periodic function, cosine manifests itself as a wave that repeats every \(2\pi\). This means the function values repeat in regular intervals along the x-axis.
For angles between \(0\) and \(\pi\), the cosine values range from \(1\) to \(-1\). Specifically, \(\cos x\) decreases from \(1\) to \(0\) as \(x\) moves from \(0\) to \(\frac{\pi}{2}\), and then from \(0\) to \(-1\) as it moves from \(\frac{\pi}{2}\) to \(\pi\).

Inverse Trigonometric Functions

Inverse trigonometric functions are the counterparts of the standard trigonometric functions for determining the angle measure from a given trigonometric ratio. For the cosine function, the inverse is the \cos^{-1}(x) or arccosine. This inverse function returns the angle whose cosine is \(x\) and is plotted within the specific range of \([0, \pi]\), ensuring the output is within an acceptable domain for standard angle measures.
For the equation \(\cos x = 0.4\), applying the inverse cosine function yields the principal value: \( x = \cos^{-1}(0.4) \), which results in an angle \(x \approx 1.16\) radians (rounded to two decimal places). This value is precise and ensures we remain within the typical constraints applied when dealing with trigonometric functions.

Interval Solutions

Finding interval solutions involves determining which angle measures satisfy the trigonometric equation within a specified range. Here, we are confined to the interval \([0, \pi]\), which is suitable for cosine because this range corresponds to the complete set of positive values of the cosine function.
Given \( \cos x = 0.4 \) within this interval, we utilize our understanding from the inverse trigonometric functions to ensure our calculated solutions, \( x = 1.16 \), fall within the defined range. Since cosine is positive in this interval, solutions will be effectively limited to the principal value derived. No other angle within this \([0, \pi]\) range satisfies the equation besides the principal angle.
Thus, the interval solutions confirm that \(x \approx 1.16\) is the sole answer that fits the criteria, providing both correctness to two decimal places and adherence to the specified interval.