Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 87

Chapter 3, Problem 87

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos x = ﹣ 2/5

Verified step by step guidance

Rewrite the equation clearly: \(\cos^2 x = -\frac{2}{5}\). This means the square of cosine of \(x\) equals a negative number.

Recall that \(\cos^2 x\) represents the square of the cosine function, which is always greater than or equal to zero for all real \(x\) because squaring any real number cannot produce a negative result.

Since \(\cos^2 x\) cannot be negative, analyze the equation \(\cos^2 x = -\frac{2}{5}\) and recognize that it has no real solutions because the right side is negative.

Conclude that there are no values of \(x\) in the interval \([0, 2\pi)\) that satisfy the equation \(\cos^2 x = -\frac{2}{5}\).

Therefore, the solution set is empty; no solutions exist for this equation within the given interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. This requires understanding the periodic nature of trigonometric functions and applying inverse functions to isolate the variable.

Using the Inverse Cosine Function

The inverse cosine function, denoted as cos⁻¹ or arccos, is used to find the angle whose cosine value is known. Since cosine is positive in the first and fourth quadrants, solutions must be considered accordingly within the interval [0, 2π).

Calculator Use and Rounding

Using a calculator to solve trigonometric equations involves inputting values correctly and interpreting results in radians or degrees as required. Rounding answers to four decimal places ensures precision and consistency in the final solutions.

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