Problem 37 - Calculus: One and Several Variables

Question

Find the number(s) \(x\) in the interval \([0.2 \pi]\) which satisfy the equation. $$\cos 2 x=0$$.

Step-by-Step Solution

Answer

The solutions to the trigonometric equation \(\cos 2x = 0\) within the interval \([0, 2 \pi]\) are \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\).

Step 1: Identify where \(\cos\) function equals zero in the unit circle

In the unit circle, cosine represents the x-coordinate. When it equals 0, we are at the top \((0,1)\) and at the bottom \((0,-1)\) of the unit circle. So cosine equals zero at angles \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).

Step 2: Apply relationship to trigonometric equation

Because we know the angles where cosine equals 0, we can write the problem as \(2x = \frac{\pi}{2}\) or \(2x = \frac{3\pi}{2}\). We are asked to find the \(x\) where \( \cos 2x = 0\).

Step 3: Solve for \(x\)

Solve the equation for x. So the possible solutions are \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\).

Step 4: Check for solutions within the given interval

As per the stipulated interval \([0, 2\pi]\), both \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\) are included, so they are our final solutions.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle centered at the origin of a coordinate plane with a radius of 1. This circle helps us understand angles and trigonometric functions visually.

• The unit circle allows us to find sine, cosine, and tangent values for various angles.
• The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, while the y-coordinate corresponds to the sine.
• Key angles such as \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) are essential as the x-coordinate is zero at these points.

For the exercise, recognizing that the cosine (x-coordinate) equals zero guides us to these specific angles for solutions. The unit circle provides a clear visual representation that simplifies understanding trigonometric identities and their applications.

Cosine Function

The cosine function is one of the main trigonometric functions and is linked closely with the unit circle. It is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

• In the unit circle, cosine is simply the x-coordinate of a point at a given angle.
• The function has a range from -1 to 1, representing all possible x-coordinates on the unit circle.
• Cosine is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

For our specific problem, we sought where \(\cos 2x = 0\). We use known points where the cosine is zero to rewrite and solve this trigonometric equation. It's all about finding where on the circle the x-value aligns with the condition described.

Interval Notation

Interval notation is a mathematical shorthand used to describe a range of values. It is often used when solving trigonometric or algebraic equations to convey the solutions that fall within a specified domain.

• It uses brackets "[" and "]" to include endpoints or parentheses "(" and ")" to exclude endpoints.
• For the problem, the interval \([0, 2\pi]\) includes all angles from 0 to \(2\pi\) radians.
• This notation is helpful in determining which solutions are valid according to given constraints.

In the exercise, ensuring that the found solutions, \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\), fit within this range is crucial. Interval notation provides a clear way to represent only those values that satisfy all conditions, ensuring we correctly capture all valid solutions.