Problem 19 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 2 \sin \theta-\sqrt{2}=0 $$

Step-by-Step Solution

Answer

\(\theta=\frac{\pi}{4}, \frac{3\pi}{4}\)

Step 1: Transpose the equation

To solve for \(\sin \theta\), transpose \(\sqrt{2}\) to the other side of the equation: \(2 \sin \theta= \sqrt{2}\)

Step 2: Solve for sine theta

Divide the equation by 2 to find the value of \(\sin \theta\): \(\sin \theta=\frac{\sqrt{2}}{2}\)

Step 3: Find possible values of theta

Since the sine function gives the same value at \( \theta\) and \( \pi- \theta\) in the first two quadrants, so we have \(\theta=\frac{\pi}{4}, \frac{3\pi}{4}\) However, \(\sin(\theta + 2\pi n) = \sin \theta\), where n is an integer. Considering the given interval for \(\theta\), \(0 \leq \theta<2 \pi\), we can ignore the effect of adding \(2\pi n\) since it is outside of this range. Hence the solutions are \(\theta=\frac{\pi}{4}, \frac{3\pi}{4}\)

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that helps us understand and measure angles. In simple terms, it relates the angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. More formally, if we have a right triangle with an angle \( \theta \), then:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

The sine function is periodic, repeating its values in a specific pattern every \(2\pi\) radians. When solving an equation like \(2 \sin \theta - \sqrt{2} = 0\), we work to isolate \(\sin \theta\) and find its possible values within a given interval, often using the properties of the sine function in a unit circle.

Unit Circle

The unit circle is a critical concept in understanding trigonometry, particularly for the sine and cosine functions. It is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle helps visualize the angle's value and how trigonometric functions behave. Here's how it works:

• The angle \( \theta \) is measured starting from the positive x-axis, moving counterclockwise.
• The coordinates of any point on the unit circle can be expressed as \((\cos \theta, \sin \theta)\).
• For \( \theta = \frac{\pi}{4} \), the point is \(\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)\).

Using the unit circle, we can see where the sine function equals specific values, helping us solve equations involving \(\sin \theta\).

Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians relate directly to the circle's radius. Here's what you need to know:

• One complete revolution around the circle equals \(2\pi\) radians.
• \(\pi\) radians is equivalent to 180 degrees.
• An angle of \(\frac{\pi}{4}\) radians corresponds to 45 degrees.

Understanding radians is crucial for working with trigonometric functions, as they are commonly used in mathematical equations and models. They offer a direct and natural way to express angles in terms of the circle's geometry.

Quadrants

The coordinate plane is divided into four quadrants, which help identify the sign and value of trigonometric functions like sine and cosine. These quadrants are:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

When solving \(\sin \theta = \frac{\sqrt{2}}{2}\), the angle \(\theta\) corresponds to values in the first and second quadrants since sine is positive in these areas. This leads to \(\theta = \frac{\pi}{4}\) in the first quadrant and \(\theta = \frac{3\pi}{4}\) in the second quadrant, helping you find solutions within the specified range of \(0 \leq \theta < 2\pi\).