Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation over the interval \([0,2 \pi)\) $$\sin \frac{x}{2}=\cos \frac{x}{2}$$

Step-by-Step Solution

Answer

\(x = \frac{\pi}{2}\) is the only solution in the interval \([0, 2\pi)\).

Step 1: Understand the problem statement

We need to solve the equation \(\sin \frac{x}{2} = \cos \frac{x}{2}\) within the interval \([0, 2\pi)\). This interval indicates the values of \(x\) that start from 0 and go up to, but do not include, \(2\pi\).

Step 2: Transform the equation

Recognize that \(\sin \theta = \cos \theta\) implies \(\theta = \frac{\pi}{4} + n\pi\) for some integer \(n\). This is because the sine and cosine functions are equal at these points.

Step 3: Apply the transformation to the equation

Substitute \(\theta = \frac{x}{2}\) into \(\theta = \frac{\pi}{4} + n\pi\) to get \(\frac{x}{2} = \frac{\pi}{4} + n\pi\).

Step 4: Solve for x

Multiply both sides of the equation \(\frac{x}{2} = \frac{\pi}{4} + n\pi\) by 2 to solve for \(x\): \(x = \frac{\pi}{2} + 2n\pi\).

Step 5: Find valid solutions in the interval

Determine the values of \(n\) such that \(0 \leq x < 2\pi\). - For \(n = 0\), \(x = \frac{\pi}{2}\).- For \(n = 1\), \(x = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2}\) (not valid since \(\frac{5\pi}{2} > 2\pi\)). - For \(n = -1\), \(x = \frac{\pi}{2} - 2\pi = -\frac{3\pi}{2}\) (not valid since negative).

Step 6: Confirm the solutions

Double-check each solution for validity within the specified interval. The only valid value for \(x\) within \([0, 2\pi)\) from the calculations above is \(x = \frac{\pi}{2}\).

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used to describe the set of all numbers between a pair of given numbers. It gives a clear way to represent which numbers belong to a set. For example, the interval \([0, 2\pi)\) includes all numbers starting from 0 up to, but not including, \(2\pi\). The bracket \("["\) indicates that the number is included in the interval while the parenthesis \(")"\) shows that the number is not included.

• \([a, b]\) means all numbers \(x\) where \(a \leq x \leq b\).
• \([a, b)\) means all numbers \(x\) where \(a \leq x < b\).
• \((a, b]\) means all numbers \(x\) where \(a < x \leq b\).
• \((a, b)\) means all numbers \(x\) where \(a < x < b\).

Understanding this notation is crucial because it precisely defines the range of valid solutions in problems, just like in trigonometric equations where solutions need to fit specific intervals.

Sine Function

The sine function is one of the fundamental functions in trigonometry. It relates the angle of a right-angled triangle to the ratio of the length of the opposite side over the hypotenuse. In a unit circle, which is a circle with a radius of 1, the sine of an angle \(\theta\) can be understood as the y-coordinate of the point where the terminal side of the angle intersects the circle.Here are some key properties of the sine function:

• It is periodic, repeating every \(2\pi\) radians.
• The range of sine is \([-1, 1]\), meaning that it will never exceed these values.
• It has specific symmetry properties, such as being an odd function, satisfying \(\sin(-\theta) = -\sin(\theta)\).

Recognizing these properties helps solve trigonometric equations, especially when they involve transformation or comparison between functions, like equating sine and cosine values.

Cosine Function

The cosine function is equally crucial in trigonometry, similar to the sine function. It defines the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Like sine, in the context of a unit circle, the cosine of an angle \(\theta\) is the x-coordinate of the point of intersection with the circle.Important aspects of the cosine function include:

• Its periodic nature, repeating every \(2\pi\) radians.
• The range of values is also constrained between \([-1, 1]\).
• It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\), which shows its symmetric property along the y-axis.

These properties become vital when we transform trigonometric equations and need to understand points where sine and cosine are equal or related in specific ways.