Problem 121 - Algebra and Trigonometry

Question

solve each equation on the interval \([0,2 \pi) .\) $$ |\cos x|=\frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

The solutions to the equation \(|\cos x|=\frac{\sqrt{3}}{2}\) on the interval \([0,2 \pi)\) are \(x = \pi/6, 5\pi/6, 7\pi/6, 11\pi/6\)

Step 1: Solve where cosine is positive

Solve the equation \(\cos x=\sqrt{3}/2\) which gives \(x = \pi/6, 11\pi/6\). This is because the cosine of both \( \pi/6 \) (or 30 degrees) and \( 11\pi/6 \) (or 330 degrees) is \( \sqrt{3}/2 \) in the interval \([0 , 2\pi)\) when cosine is positive (i.e., on the intervals \([0, \pi/2]\) and \([3\pi/2 , 2\pi)\)).

Step 2: Solve where cosine is negative

Solve the equation \( \cos x=-\sqrt{3}/2\), which gives \(x = 5\pi/6, 7\pi/6\). This is because the cosine of both \( 5\pi/6 \) (or 150 degrees) and \( 7\pi/6 \) (or 210 degrees) is \( -\sqrt{3}/2 \) in the interval \([0 , 2\pi)\) when cosine is negative (i.e., on the intervals \((\pi/2 , 3\pi/2)\)).

Key Concepts

Cosine Function

The cosine function is a fundamental element in trigonometry, which relates the angles in a right triangle to the ratio of the adjacent side over the hypotenuse. The function itself, denoted as \( \cos \), is periodic and repeats every \(2\pi\) radians or 360 degrees. This property makes it especially useful for solving problems involving circular motion and oscillations.

When it comes to the unit circle, the cosine of an angle \(x\) represents the x-coordinate of a point on the circumference of the circle. This interpretation aids in solving equations where the cosine is set equal to a particular value, as in the given exercise. It's important to remember that \( \cos(\pi/6) \approx \cos(11\pi/6) = \sqrt{3}/2 \) and \( \cos(5\pi/6) \approx \cos(7\pi/6) = -\sqrt{3}/2 \). These values result from the symmetry and periodicity of the cosine function.

Absolute Value Equations

An absolute value equation involves the absolute value function, which outputs the non-negative magnitude of a given real number or expression, regardless of its sign. The notation \(|a|\) denotes the absolute value of \(a\). In the context of the exercise, the equation \(|\cos x| = \frac{\sqrt{3}}{2}\) implies that \(\cos x\) itself can be either \(\frac{\sqrt{3}}{2}\) or \( -\frac{\sqrt{3}}{2}\), as the absolute value removes any negative sign.

In solving absolute value equations, one typically separates the problem into two cases based on the positivity or negativity of the inside expression. This approach helps to capture all the possible angles that satisfy the original absolute value condition, leading to the identification of the correct solutions within the specified interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any value of the variable. Common identities include the Pythagorean identities, angle sum and difference formulas, double angle formulas, and others. These identities are immensely useful when simplifying trigonometric expressions, proving other mathematical results, or solving trigonometric equations.

In the exercise provided, knowledge of the basic values of the cosine function and the periodic properties of trigonometric functions are utilized. It is a straightforward application of trigonometric identities, rather than more complex identities, yet understanding the core principle behind these identities is crucial for students to solve a wide array of trigonometry problems effectively.