Question

In Exercises \(31-36,\) solve the equation in the specified interval. $$\sin x=-0.5\( \)-\infty\( \)< x <\( \)\infty$$

Short answer

The solutions are \( x = 7\pi /6 + 2\pi n \) and \( x = 11\pi /6 + 2\pi n \), where n is any integer.

Step-by-step solution

Understand the sine function

The sine function, \( \sin x \), gives us values from -1 to 1 based on the y-coordinate in the unit circle. When the sine of an angle is -0.5, this corresponds to values at \(7\pi /6 \) and \(11\pi /6 \) in the unit circle.

Solve for x

In the interval \(-\infty < x < \infty\), there are multiple solutions. We have two base solutions from the unit circle (Step 1), but sine has a period of \(2\pi\). Therefore, we can add \(2\pi * n\) to each base solution to get all possible solutions. Thus, \( x = 7\pi /6 + 2\pi n \) and \( x = 11\pi /6 + 2\pi n \), where n is an integer.

Write the final solutions

The solutions to the equation \( \sin x = -0.5 \) in the interval \(-\infty < x < \infty\) are \( x = 7\pi /6 + 2\pi n \) and \( x = 11\pi /6 + 2\pi n \) where n is an integer.

Key concepts

Sine Function

The sine function is a fundamental trigonometric function that appears frequently in mathematics, particularly in the study of waves, oscillations, and circular motion. It relates an angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, which is a circle with a radius of one, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

Understanding the sine function is vital when solving trigonometric equations. In the given exercise, the equation to be solved is \( \sin x = -0.5 \). Angles that have this particular sine value correspond to specific positions on the unit circle, which leads to the identification of the base solutions. It’s important to note that these angles are in radians, a measure that will be elaborated on in a following section.

Unit Circle

The unit circle is an essential concept in trigonometry, defining a circle with a radius of one unit. It's centered at the origin of a coordinate plane, and is used to define trigonometric functions. The circle provides a geometric representation for angles and their corresponding trigonometric values.

In our exercise, the unit circle helps us identify the angles whose sine value is -0.5. Generally, a complete rotation around the unit circle represents an angle of \( 2\pi \) radians. By using the circle, we can see that the angles producing a sine of -0.5 lie in the third and fourth quadrants, specifically at \( 7\pi/6 \) and \( 11\pi/6 \).

Every point on the unit circle corresponds to an angle and its sine and cosine values. Therefore, understanding the unit circle is crucial for solving trigonometric equations, as it allows you to visualize the relationship between angles and their sine or cosine values.

Periodicity of Trigonometric Functions

Trigonometric functions such as sine and cosine are periodic, which means they repeat their values in regular intervals. For the sine function, this interval is \( 2\pi \) radians, which corresponds to a full rotation around the unit circle. This periodicity is a key concept when solving trigonometric equations because it means that there can be infinitely many solutions that follow a pattern.

In the given problem, we use the periodicity of the sine function to find all possible solutions within the interval \( -\infty < x < \infty \). Starting from the base solutions \( 7\pi/6 \) and \( 11\pi/6 \), we can add multiples of the period \( 2\pi \) (where n is an integer) to find all the angles that have a sine of -0.5. Thus, the understanding of periodicity allows students to generalize the solutions and ensures they can find all possible angles that satisfy the equation.

Radians

Radians are a unit of measure for angles based on the radius of a circle. In trigonometry, the radian measure is preferred over degrees because it simplifies many formulas and calculations. A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

There are approximately 6.28318 (or \( 2\pi \) in terms of \(\pi\)) radians in a full circle, which is equivalent to 360 degrees. This means that \(\pi\) radians are equal to 180 degrees. Using radians allows the use of the unit circle and promotes a more natural understanding of the periodic nature of trigonometric functions.

By using radians, we can describe the solutions for our trigonometric equation in a more mathematically elegant and precise manner, such as \( x = 7\pi/6 + 2\pi n \) and \( x = 11\pi/6 + 2\pi n \) where \(n\) is an integer, covering all the possible solutions within the infinite interval.