Problem 155

Question

For the following exercises, solve the trigonometric equations on the interval $0 \leq \theta<2 \pi.$ $$2 \sin \theta-1=0$$

Step-by-Step Solution

Verified

Answer

$\theta = \frac{\pi}{6}, \frac{5\pi}{6}$.

1Step 1: Isolate the Trigonometric Function

The given equation is $2 \sin \theta - 1 = 0$. First, we need to isolate the sine function. To do this, add 1 to both sides of the equation, resulting in $2 \sin \theta = 1$.

2Step 2: Solve for Sine

Now, divide both sides of the equation by 2 to solve for $\sin \theta$. This gives us $\sin \theta = \frac{1}{2}$.

3Step 3: Find General Solutions

Using the unit circle, we know that $\sin \theta = \frac{1}{2}$ at $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$ within one cycle of the circle.

4Step 4: Verify Interval

The problem specifies that we need solutions in the interval $0 \leq \theta < 2\pi$. Both $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ satisfy this constraint, so they are valid solutions.

Key Concepts

Solve for SineUnit CircleInterval [0, 2π]

Solve for Sine

To solve a trigonometric equation like this one, you'll often aim to isolate the trigonometric function on one side of the equation. Here, we start with the equation $2 \sin \theta - 1 = 0$. Our target is to get $\sin \theta$ by itself.

First, add 1 to both sides to get it closer to the sine term: $2 \sin \theta = 1$.
Next, divide both sides by 2, achieving $\sin \theta = \frac{1}{2}$.

This isolation of $\sin \theta$ allows us to move on to the next step: finding the angle $\theta$ which corresponds to this sine value.

Unit Circle

The unit circle is a foundational concept in trigonometry. It's a circle of radius one, centered at the origin of a coordinate system. The unique property of the unit circle is that the coordinates of each point on the circle can describe sine and cosine values for angles.

The sine of an angle $\theta$ is the y-coordinate of the point where the radius intersects the unit circle.
When we need to find $\theta$ for which $\sin \theta = \frac{1}{2}$, we look at the unit circle.
Common angles with known sine values can quickly help identify solutions. For $\sin \theta = \frac{1}{2}$, these angles are $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$.

Referencing the unit circle is crucial because it provides the exact points/angles where the sine (or cosine) values can occur naturally without need for a calculator.

Interval [0, 2π]

The interval $[0, 2\pi)$ is important because it defines the range we're interested in finding solutions for $\theta$. This interval corresponds to one full revolution around the unit circle (360 degrees).

$0$ and $2\pi$ correspond to the same point on the circle, so $2\pi$ is often excluded, hence the notation $[0, 2\pi)$.
Common task in trigonometry problems is verifying if the calculated solutions (e.g., $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$ in this problem) fit within this specific interval.
Ensuring solutions fall within this interval completes the exercise, confirming that these angles are valid and acceptable answers to the problem.

Solutions not within the given interval would require additional considerations or adjustments to fit the defined range.

Problem 154

Problem 155

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Problem 154

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Problem 154

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Problem 156

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