Problem 71 - Algebra and Trigonometry

Question

\(71-74\). 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 approximately \(x = 1.16\).

Step 1: Understanding the Problem

We are given the equation \(\cos x = 0.4\) and need to find the solutions for \(x\) that lie in the interval \([0, \pi]\).

Step 2: Find the Principal Solution

The principal solution for \(x\) when \(\cos x = 0.4\) can be found using the inverse cosine function: \(x = \cos^{-1}(0.4)\). Using a calculator, compute this value and round it to two decimal places.

Step 3: Calculate the Principal Value

Using a calculator or a cosine table, \(\cos^{-1}(0.4) \approx 1.16\) radians. This is the principal value of \(x\) that satisfies the equation in the given interval.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are a way of finding the angle when given a trigonometric ratio. For example, if you know \( \cos x = 0.4\\), but you need to find the angle \(x\), you use the inverse cosine function, written as \(\cos^{-1}(0.4)\). This tells us the angle whose cosine is 0.4. In terms of function definitions, inverse functions reverse the operations of their respective trigonometric functions. It is helpful because it provides an angle as the output rather than a ratio. Consider the calculator function usually marked as \(\cos^{-1}\) or "acos", which allows you to enter your cosine value and compute the angle.

• Inverse functions only return a principal value, which is the most common angle that lies within a specific range.
• For cosine, the principal value lies in the interval \(0≤x≤π\).

Understanding inverse trigonometric functions is important when solving problems that require determining angle measurements given a trigonometric ratio.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the length of the adjacent side over the hypotenuse. Specifically, for an angle \(\theta\), the cosine is defined as: \[\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]The value of cosine ranges between -1 and 1. In the unit circle representation, the cosine of an angle corresponds to the x-coordinate of the point on the unit circle, where the angle is measured from the positive x-axis.

• The function is periodic, repeating every \(2\pi\) radians, which means any solution in one period maps to similar solutions in other periods.
• Cosine is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).

In the exercise, we specifically looked for the angle satisfying \(\cos x = 0.4\). By using the inverse function, we found the angle in the interval \[0, \pi\].

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. It is one of the two major units of measuring angles, the other being degrees. In the context of trigonometry and calculus, radians are often more convenient.

• One radian is defined as the angle created when the arc length is equal to the radius of the circle.
• There are \(2\pi\) radians in a complete circle compared to 360 degrees, making \(\pi\) radians equal to 180 degrees.

Using radians allows for more straightforward mathematical formulations, especially in calculus, because the derivative and integral formulas of trigonometric functions become much simpler. In the exercise, calculating \(\cos^{-1}(0.4)\) yielded a solution in radians (approximately 1.16), showing how radians naturally fit into solving trigonometric problems. Understanding radian measure is essential for interpreting angles and solutions in trigonometric equations.