Problem 87 - Precalculus Essentials

Question

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\cos x=-\frac{2}{5}$$

Step-by-Step Solution

Answer

The solution to the equation \(\cos x= -\frac{2}{5}\) in the interval \([0,2\pi)\) is \(x= 2\pi - \arccos -\frac{2}{5}\), rounded to four decimal places.

Step 1: Solve for \(x\) using inverse cosine

Using a scientific calculator, calculate the value of \(x\) using the inverse cosine or arccos function. For \(\cos x= -\frac{2}{5}\), we get \(x=\arccos -\frac{2}{5}\).

Step 2: Adjust the answer

The calculator will provide a value in the range \([0,\pi]\). As the cosine is negative, the answer is actually in the second quadrant. Therefore, subtract the calculator's value from \(2\pi\) to get the correct value of \(x\). So we get \(x= 2\pi - \arccos -\frac{2}{5}\).

Step 3: Precision

Ensure the answer is correct to four decimal places as instructed.

Key Concepts

Understanding Inverse Cosine

When solving trigonometric equations, the inverse cosine function, often denoted as \( \cos^{-1} \) or \( \arccos \), is a vital tool. It helps us find an angle whose cosine value we already know. For example, if we're given \( \cos x = -\frac{2}{5} \), using the inverse cosine lets us determine the angle \( x \) corresponding to this cosine value.

The inverse cosine function is limited to providing results within the principal range of \([0,\pi]\). This is important because cosine covers all possible values from \(-1\) to \(1\) within this range. In practice, this means when you use a calculator to compute \( \arccos(-\frac{2}{5}) \), it will return the angle in this principal range.

However, keep in mind that even though your calculator provides one specific angle, other angles with the same cosine value may exist. That's where understanding the location of angles in different quadrants comes into play.

Using a Scientific Calculator

A scientific calculator is a powerful tool for tackling trigonometric equations efficiently. Most modern calculators are equipped with a function to calculate the inverse cosine, usually labeled as \( \cos^{-1} \) or \( \arccos \). Here's a simple guide on how to use it:

• Enter the value, such as \(-\frac{2}{5}\), into your calculator.
• Press the button designated for inverse cosine.
• Read off the angle, usually displayed in radians if your calculator is set accordingly.

Make sure your calculator is set to the correct mode based on the unit system used in your exercise, either degrees or radians. For this problem, since the intervals are in radians (\([0, 2\pi)\)), ensure it is set to radians.

Scientific calculators provide accurate answers but remember that rounding is crucial. Solutions often require results to a specific number of decimal places, such as four in this instance. Adjust your number display settings if needed to match the required precision.

Exploring Quadrants in Trigonometry

Understanding the quadrants is crucial when solving trigonometric equations, especially when angles aren't restricted to the first quadrant. The four quadrants divide the coordinate plane and influence the sign and value of trigonometric functions.

• First Quadrant (0 to \(\pi/2\)): All trigonometric functions, including cosine, are positive here.
• Second Quadrant (\(\pi/2\) to \(\pi\)): Cosine turns negative. Angles computed by \(\arccos\) for negative values fall within this quadrant.
• Third Quadrant (\(\pi\) to \(3\pi/2\)): Here, both sine and cosine are negative.
• Fourth Quadrant (\(3\pi/2\) to \(2\pi\)): Cosine becomes positive again, but sine remains negative.

When given \(\cos x = -\frac{2}{5}\) in our problem, it first suggests a position in the second quadrant since cosine is negative. However, knowing the periodic nature of cosine, we also find that cosine can be negative again in the third quadrant, and thus, you may need to adjust the angle provided by the calculator using the formulas explained earlier, especially when dealing with broader intervals like \([0, 2\pi)\).