Problem 70 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\sin \theta=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

The possible values of \(\theta\) are \(60^\circ\) and \(120^\circ\).

Step 1: Understand the range of sine function

The sine function, \(\sin \theta\), has a range of values from -1 to 1. Therefore, the value \(\frac{\sqrt{3}}{2}\) is within this range, indicating there are solutions for \(\theta\).

Step 2: Identify the known angles

Recall that \(\sin 60^\circ = \frac{\sqrt{3}}{2}\). Therefore, \(\theta = 60^\circ\) is one possible solution.

Step 3: Consider the unit circle property

The sine function is positive in the first and second quadrants. This means there may be another angle in the second quadrant where \(\sin \theta = \frac{\sqrt{3}}{2}\).

Step 4: Determine the second solution in the second quadrant

In the second quadrant, \(\sin(180^\circ - \theta) = \sin \theta\). Setting \(180^\circ - \theta = 60^\circ\), we solve for \(\theta\): \(\theta = 180^\circ - 60^\circ = 120^\circ\).

Step 5: Verify the solutions

Verify that both \(60^\circ\) and \(120^\circ\) are within the required range and that \(\sin 60^\circ = \sin 120^\circ = \frac{\sqrt{3}}{2}\).

Step 6: List all the solutions

The possible values of \(\theta\) in the interval \(0^\circ \leq \theta \leq 360^\circ\) are \(60^\circ\) and \(120^\circ\).

Key Concepts

The Sine Function

The sine function is one of the basic trigonometric functions, commonly denoted as \( \sin \theta \). It relates the angle \( \theta \) in a right-angled triangle to the ratio of the length of the opposite side over the hypotenuse. This function is periodic and repeats every \( 360^{\circ} \) or \( 2\pi \) radians.
The sine function values range between -1 and 1, making it suitable for describing oscillations, such as sound waves or alternating current. For example, in the range [0, 360°], \( \sin 60^{\circ} \) gives a positive value of \( \frac{\sqrt{3}}{2} \).
Importantly, due to the nature of sine, if you have one positive value, like \( \frac{\sqrt{3}}{2} \), there may be multiple angle solutions within a full rotation, as seen in this exercise.

The Unit Circle

The unit circle is a powerful tool in trigonometry, helping us visualize angles and their corresponding sine and cosine values. The circle is centered at the origin of a coordinate plane with a radius of 1 unit. Each point on the circle can be described by an angle \( \theta \) from the positive x-axis, measured counterclockwise.
Because the radius is 1, any point on the circle given by coordinates (x, y) will have a connection to sine and cosine. Specifically, \( x = \cos \theta \) and \( y = \sin \theta \). So, for an angle like \( \theta = 60^{\circ} \), the point on the circle translates to \( (\frac{1}{2}, \frac{\sqrt{3}}{2}) \).
Moving around the unit circle also helps us determine angles in different quadrants based on where the line intersects the circle. This becomes especially useful when identifying where the sine function is positive or negative.

Understanding Angle Measures

When dealing with trigonometric equations, understanding angle measures is key. Angles can be measured in degrees or radians, but in many math courses, degrees are more intuitive. An angle can open clockwise or counterclockwise from the positive x-axis.
For example, the angles \( 60^{\circ} \) and \( 120^{\circ} \) in this exercise show how different angles can yield the same sine value.\( 60^{\circ} \) measures a short arc in the first quadrant, while \( 120^{\circ} \) is the equivalent mirror in the second quadrant.
These conversions and references are crucial when solving trigonometric problems, helping you translate abstract math into concrete visual references.

Exploring Quadrants

The coordinate plane is divided into four quadrants, each providing valuable insight into how trigonometric functions like sine behave. Starting in the upper right and moving counterclockwise, the 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.

This distinction matters because it determines the sign of the trigonometric outputs. In this exercise, the sine function's value is positive, which implies that the angle solutions \( \theta \) reside in the first and second quadrants.
Understanding these properties helps in identifying other possible angle solutions for a given sine value and explains why \( 60^{\circ} \) and \( 120^{\circ} \) fit the criteria for \( \sin \theta = \frac{\sqrt{3}}{2} \).