Problem 41 - Calculus: One and Several Variables

Question

Find the number \((s) x\) in the interval \([0,2 \pi j]\) which satisfy the equation. $$\sin x=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The solutions of the equation \(\sin x = -\frac{1}{2}\) within the interval \([0 , 2\pi]\) are \(x = 7\pi/6\) and \(x = 11\pi/6\).

Step 1: Determine the Quadrants

To find the values of \(x\) such that \(\sin x = -\frac{1}{2}\), it's important to know that sine is negative in the third and fourth quadrants of the unit circle.

Step 2: Find the standard angle

We should find the reference (or 'standard') angle which has sine value of \(+1/2\) in first or second quadrant. That angle is \(x=\pi/6\) or \(x=30^\circ\) in the first quadrant.

Step 3: Apply to negative sine

Sine is negative in the third and the fourth quadrants. In the third quadrant it's \(\pi + \pi/6 = 7\pi/6\). In fourth quadrant, it's \(2\pi - \pi/6 = 11\pi/6\)

Step 4: Verify within boundaries

Finally, it's important to verify that these solutions fall within the given boundary of \([0 , 2\pi]\). Both \(7\pi/6\) and \(11\pi/6\) do.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. This circle helps in finding the sine, cosine, and tangent of angles, which are based on the coordinates of points where the circle intersects with terminal sides of the angles.

• The unit circle allows us to visualize and understand the behavior of trigonometric functions over a complete revolution.
• The angle values on the unit circle typically range from 0 to \(2\pi\) radians or 0 to 360 degrees, completing one full circle.

In the case of the sine function, the y-coordinate of the point on the unit circle determines the value of sine for a given angle.
Therefore, for any point \( (x, y) \) on the unit circle, the sine of the angle is equal to the y-coordinate of the point.

Quadrants

In the context of the unit circle, the coordinate plane is divided into four quadrants. Each quadrant represents a section of the circle where the signs of the sine and cosine values differ.

• First Quadrant: Both sine and cosine values are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Sine is negative, cosine is negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

Knowing in which quadrant an angle lies is useful for determining the sign of the trigonometric function values. For example, since sine is negative in the third and fourth quadrants, the solutions to \( \sin x = -\frac{1}{2} \) must be found in these quadrants.

Reference Angle

A reference angle is the acute angle that a given angle makes with the x-axis. It helps in determining the value of trigonometric functions for angles that lie outside the first quadrant.

• Reference angles are always positive and are typically less than \( \pi/2\) or 90 degrees.
• They are used to find the equivalent value of a trigonometric function in another quadrant.

In the problem of finding \( \sin x = -\frac{1}{2} \), the reference angle is \( \pi/6 \) or 30 degrees. This is because \( \sin(\pi/6) = \frac{1}{2} \), and we use this reference angle to determine the sine of angles in the third and fourth quadrants, which are \( 7\pi/6 \) and \( 11\pi/6 \).

Sine Function

The sine function, one of the primary trigonometric functions, is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle.
However, when using the unit circle, it is defined as the y-coordinate of the point on the circle corresponding to an angle.

• The sine function is periodic with a period of \( 2\pi \) radians (or 360 degrees).
• It begins at 0, reaches a maximum of 1 at \( \pi/2 \), returns to 0 at \( \pi \), reaches -1 at \( 3\pi/2 \), and completes a cycle back at 0 at \( 2\pi \).

This cyclical behavior helps in solving equations like \( \sin x = -\frac{1}{2} \) by identifying the angle positions on the unit circle and applying the correct signs based on the relevant quadrants.