Problem 34 - Precalculus

Question

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi), f(x)=\frac{\sqrt{2}}{2} .\) Find all values of \(x\)

Step-by-Step Solution

Answer

The values of \(x\) are \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\).

Step 1: Understand the problem

We need to find all the values of \(x\) in the interval \([0, 2\pi)\) where \(f(x) = \sin x = \frac{\sqrt{2}}{2}\).

Step 2: Recall the sine function values

Recall the unit circle values where \(\sin x = \frac{\sqrt{2}}{2}\). These values are at angles \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\) in one full rotation \([0, 2\pi)\).

Step 3: Identify values in given interval

Given that the domain is \([0, 2\pi)\), check whether both \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\) fall within this interval. In this case, both values are within \([0, 2\pi)\).

Step 4: Verify the solution

Double-check by plugging the values \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\) back into \(f(x) = \sin x\) to ensure they both result in \(\frac{\sqrt{2}}{2}\). Indeed, \(\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) and \(\sin \left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 5: Conclusion

Therefore, the solutions for \(x\) are \[ \frac{\pi}{4}, \frac{3\pi}{4} \].

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used in mathematics. It arises frequently in geometry, physics, engineering, and other fields requiring the study of waves and periodic phenomena. The function is defined for any angle in radians and is written as \(\sin x\). The sine function measures the vertical component of a point on the unit circle. It outputs the y-coordinate of that point, which lies on a circle with a radius of 1.
Some key characteristics of the sine function include:

• Periodic behavior with a cycle of \(2\pi\). This means the sine function repeats its values every \(2\pi\) radians.
• An amplitude of 1, indicating its maximum and minimum outputs are 1 and -1, respectively.
• The sine function is an odd function, showing symmetry about the origin: \(\sin(-x) = -\sin x\).

When solving equations involving \(\sin x\), such as the exercise \(f(x) = \sin x = \frac{\sqrt{2}}{2}\), identifying known sine values and corresponding angles like \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\) is crucial.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a valuable tool in trigonometry because it allows us to easily determine the sine and cosine values for various angles. Every point on the unit circle can be represented as \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle measured from the positive x-axis.
Here are important points about the unit circle:

• It provides a geometric representation of trigonometric functions. The x-coordinate of a point is the cosine of the angle, and the y-coordinate is the sine of the angle.
• Significant angles and their sine and cosine values are often memorized using the unit circle, such as \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), and \(\pi\).
• The unit circle makes it easier to visualize and solve trigonometric equations, like finding where \(\sin x = \frac{\sqrt{2}}{2}\), by identifying points on the circle corresponding to these values.

The unit circle is especially helpful for understanding the cyclical nature of trig functions and finding solutions over specific intervals.

Angles in Radians

Angles in trigonometry are often measured in radians, a unit and method of measuring angles based on the radius of a circle. One full circle rotation is equal to \(2\pi\) radians, which equates to 360 degrees. Converting between radians and degrees can be helpful in many mathematical applications.
Here are some important points about radians:

• In the radian system, the angle is calculated by taking the arc length divided by the radius of the circle.
• Common angles in radians include: \(\pi/6\), \(\pi/4\), \(\pi/3\), and \(\pi/2\), corresponding to 30°, 45°, 60°, and 90°, respectively.
• Radians allow the trigonometric functions to be periodic with a simpler period of \(2\pi\), easing calculations in calculus and physics.

Understanding radians and being able to switch between radians and degrees is a fundamental skill in trigonometry, crucial for interpreting problems and solutions in different mathematical contexts.