Problem 9 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 3 \sin \theta-\sqrt{3}=\sin \theta $$

Step-by-Step Solution

Answer

The exact values for \( \theta \) are \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \).

Step 1: Simplify the Equation

Start by simplifying the equation given: \( 3 \sin \theta - \sqrt{3} = \sin \theta \). Subtract \( \sin \theta \) from both sides to get: \( 2 \sin \theta = \sqrt{3} \).

Step 2: Solve for \( \sin \theta \)

Divide both sides of the equation by 2 to solve for \( \sin \theta \): \( \sin \theta = \frac{\sqrt{3}}{2} \).

Step 3: Find General Solutions for \( \sin \theta = \frac{\sqrt{3}}{2} \)

Recall that \( \sin \theta = \frac{\sqrt{3}}{2} \) at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \). These are the angles in one cycle (0 to \( \pi \)) where the sine reaches \( \frac{\sqrt{3}}{2} \).

Step 4: Verify the Interval

Ensure these solutions lie within the interval \( 0 \leq \theta < 2\pi \). Both \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \) fit within this interval.

Key Concepts

Interval Notation

Interval notation is a mathematical method to express a range of values. For instance, the interval \(0 \leq \theta < 2\pi\) indicates that \(\theta\) can take any value starting from 0 up to (but not including) \(2\pi\).

Understanding this is crucial when solving trigonometric equations, as it helps to specify the solutions within a particular range. In this exercise, we're asked to find solutions of \(\theta\) in this interval, ensuring our final answers are valid within the bounds given.

Here's how to interpret the different symbols:

• \([ \) or \( ]\) - Indicates the interval is closed on that side, meaning it includes the endpoint.
• \(( \) or \()\) - Indicates the interval is open on that side, meaning it does not include the endpoint.

Understanding these distinctions ensures you provide all possible valid solutions for your trigonometric equations within the specified range.

Sine Function

The sine function, one of the primary trigonometric functions, measures the ratio of the opposite side to the hypotenuse in a right triangle.

In terms of the unit circle, it represents the y-coordinate of a point corresponding to an angle \(\theta\). The values of \(\sin \theta\) repeat every \(2\pi\), which is the period of the sine function, meaning that the function's value completes a full cycle as \(\theta\) ranges from 0 to \(2\pi\).

In this exercise, you encountered the equation \(3 \sin \theta - \sqrt{3} = \sin \theta\). Simplifying it, you solved for \(\sin \theta\), showing that understanding the sine function's properties can simplify complex-looking equations.

Key Points about Sine Function:

• Range: [-1, 1]
• Periodicity: \(2\pi\)
• Important angles: \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)

These points help in solving equations and understanding the behavior of the sine curve over one complete cycle.

Exact Trigonometric Values

Exact trigonometric values are specific angles at which trigonometric functions yield precise results. These values come from well-known angles commonly used in mathematical problems.

Common Exact Values:

• \(\sin \frac{\pi}{6} = \frac{1}{2}\), \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)
• \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\), \(\cos \frac{\pi}{3} = \frac{1}{2}\)

In this exercise, solving the equation led to \(\sin \theta = \frac{\sqrt{3}}{2}\). By recalling precise trigonometric values, you identified the angles \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\).

These angles are crucial as they help confirm which solutions fit perfectly within the given interval. Memorizing these values helps solve trigonometric equations faster and more accurately.

Unit Circle

The unit circle is a vital tool in understanding trigonometry. It is a circle with a radius of one centered at the origin of a coordinate plane.

The unit circle helps visualize and determine the values of trigonometric functions for different angles. As \(\theta\) varies, the coordinates \((x, y)\) on the circle's circumference define the values of \(\cos \theta = x\) and \(\sin \theta = y\).

Why Use the Unit Circle?

• It provides a geometric interpretation of trigonometric functions.
• Helps easily identify exact trigonometric values.
• Offers a comprehensive view of all angle measures within \(0\) to \(2\pi\).

In this exercise, once you determined \(\sin \theta = \frac{\sqrt{3}}{2}\), the unit circle quickly showed the angles at which this happens.

Using the unit circle simplifies verifying the interval condition \(0 \leq \theta < 2\pi\) as well, making it an indispensable tool in your trigonometry toolkit.