Problem 25 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 2 \sin \theta=-\sqrt{3} $$

Step-by-Step Solution

Answer

The solutions to the given equation in the range \(0 \leq \theta < 2 \pi\) are \(\theta = \frac{4\pi}{3}\) and \(\theta = \frac{5\pi}{3}\).

Step 1: Isolate the sin function

We start by isolating the sin function on one side of the equation. To achieve this, we will divide both sides of the equation by 2, which modifies the equation to: \[\sin \theta = -\frac{\sqrt{3}}{2}\]

Step 2: Determine the angle

Now, we can see that \( \sin \theta \) equals some negative number. Referring to the unit circle and recognizing common angles, we can recall that \(\sin \theta = -\frac{\sqrt{3}}{2}\) corresponds to \(\theta = \frac{4\pi}{3}\) and \(\theta = \frac{5\pi}{3}\).

Step 3: Verifying the solution

These values are the solutions because they fall within the specified range \(0 \leq \theta < 2 \pi\) and obtain when \(\sin \theta = -\frac{\sqrt{3}}{2}\). We could verify these values by substituting back into the original equation for \(\theta\).

Key Concepts

Unit Circle

The unit circle is a key concept in trigonometry. It includes a circle with a radius of 1 and is centered at the origin of a coordinate plane. This tool is pivotal because it helps us find angles and solve trigonometric functions like sine, cosine, and tangent easily.

• The circle's radius is always 1 unit.
• The coordinates of any point on the unit circle can be written as \((\cos \theta, \sin \theta)\).
• It covers angles from 0 to \(2\pi\) radians (or 0 to 360 degrees).

The angles on the unit circle are measured in radians, where \(2\pi\) radians equate to a full circle. Moreover, each quadrant, moving counter-clockwise from the positive x-axis, spans \(\frac{\pi}{2}\) radians. Here, negative angles and negative sine or cosine values can be easily identified by their respective positions and symmetries across the circle.

Sine Function

The sine function is one of the primary trigonometric functions used to describe the relationship between the angles and sides of right-angle triangles. It's frequently represented in terms of the unit circle, where the y-coordinate of any point gives the value of the sine function for that angle.

• Sine values range between -1 and 1.
• When measured from the unit circle, sine represents vertical displacement.
• Key angles to remember where sine values are simple to recall include \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), etc.

In this context, the function \( \sin \theta = -\frac{\sqrt{3}}{2} \) specifically relies on identifying angles where the sine of the angle is negative. This occurs in the third and fourth quadrants of the unit circle.

Trigonometric Solutions

When solving trigonometric equations, understanding the unit circle and sine function is crucial. These solutions revolve around determining the angles that satisfy given trigonometric expressions.
For the equation \(2 \sin \theta = -\sqrt{3}\):

• We first isolate the sine function, determining \( \sin \theta = -\frac{\sqrt{3}}{2} \).
• Refer to familiar angles on the unit circle to find which angles produce this sine value. Knowing common angles and their sines from the unit circle makes this easier.
• The solutions \(\theta = \frac{4\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) provide the required sine values within the interval \(0 \leq \theta < 2\pi\).

Verification ensures accuracy. By substituting the solutions back into the original equation, you confirm they fit within the given range and satisfy the trigonometric equation.