Problem 52 - Precalculus with Calculus Previews

Question

Find all angles \(\theta\), where \(0^{\circ} \leq \theta<\) \(360^{\circ}\), that satisfy the given condition. $$ \sin \theta=-\frac{1}{2} $$

Step-by-Step Solution

Answer

The angles are \(\theta = 210^{\circ}\) and \(\theta = 330^{\circ}\).

Step 1: Understand the Sine Function Range

The sine function, \(\sin \theta\), has values ranging from -1 to 1. We are given \(\sin \theta = -\frac{1}{2}\). This value is within the possible range, so solutions exist.

Step 2: Know Sine's Special Angles

We need angles where the sine of the angle is \(-\frac{1}{2}\). This corresponds to angles of \(30^{\circ}\) and \(150^{\circ}\) in standard positions but negative in value, which occurs in the third and fourth quadrants for negative sine values.

Step 3: Determine Reference Angle

The reference angle for \(\sin \theta = \frac{1}{2}\) is \(30^{\circ}\). For negative values, \(\theta\) will be in either the third quadrant or the fourth quadrant.

Step 4: Calculate Angles in the Third Quadrant

In the third quadrant, the sine function is negative. The angle that corresponds to \(30^{\circ}\) is \(\theta = 180^{\circ} + 30^{\circ} = 210^{\circ}\).

Step 5: Calculate Angles in the Fourth Quadrant

In the fourth quadrant, the sine function is also negative. The angle that corresponds to \(30^{\circ}\) is \(\theta = 360^{\circ} - 30^{\circ} = 330^{\circ}\).

Step 6: Verify Solutions

We calculate \(\sin 210^{\circ}\) and \(\sin 330^{\circ}\) to verify they equal \(-\frac{1}{2}\). Both satisfy the original equation, confirming the solutions are correct.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often represented as \( \sin \theta \). It relates to the Y-coordinate of a point on the unit circle that corresponds to the angle \( \theta \). The sine function has a range of values between -1 and 1. This means that for any angle \( \theta \), \( \sin \theta \) will always lie within this interval.

Understanding the range is crucial when solving trigonometric equations. For example, the problem we are tackling involves finding angles where \( \sin \theta = -\frac{1}{2} \). Knowing the range of sine ensures us that such a value is valid. Recognizing this range is a foundational skill in trigonometry, allowing you to determine when a solution is possible or if an error may exist in the problem setup.

• Sine function range: -1 to 1
• Corresponds to Y-coordinates on the unit circle

Reference Angle

A reference angle is a crucial concept when working with trig functions. It is the smallest angle that the terminal side of a given angle makes with the X-axis. Reference angles are always positive and less than \(90^\circ\).

For \( \sin \theta = -\frac{1}{2} \), we need to identify the corresponding positive angle on the unit circle. This is where the reference angle comes into play. The reference angle for a sine value of \( \frac{1}{2} \) corresponds to \( 30^\circ \). Finding this first helps determine in which quadrants the negative sine value occurs. In this case, the sine value will be negative, aligning with our original equation in certain quadrants.

• Reference angles are always positive, under \(90^\circ\)
• Helps find equivalent angles across different quadrants

Quadrants

When dealing with trigonometric functions, the position of an angle in the coordinate plane is defined by quadrants. The plane is divided into four quadrants, each influencing the sign of trigonometric functions:

• 1st Quadrant: All trigonometric functions are positive.
• 2nd Quadrant: Only sine is positive; others are negative.
• 3rd Quadrant: Tangent is positive; sine and cosine are negative.
• 4th Quadrant: Cosine is positive; sine and tangent are negative.

For the exercise \( \sin \theta = -\frac{1}{2} \), the interest lies in the 3rd and 4th quadrants. In the 3rd quadrant, angles are calculated as \( 180^\circ + \text{reference angle} \), and in the 4th, angles are \( 360^\circ - \text{reference angle} \). Hence, the angles obtained are \( 210^\circ \) and \( 330^\circ \), respectively.

Understanding the behavior of sine in each quadrant allows you to precisely calculate the desired angles and confirm their validity, ensuring they produce the correct negative sine value as needed.