Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos x=-\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

The solutions to the equation in the interval \([0,2\pi)\) are \(5\pi/6\) and \(7\pi/6\).

Step 1: Understand the properties of cosine function

First, understand that the cosine function, denoted as cos(x), gives us the x-coordinate of the point on the unit circle determined by the angle x. Moreover, cosine function is positive in the first and fourth quadrants, and negative in the second and third quadrants of the unit circle.

Step 2: Determine the reference angles

The given value of cos(x) is \(-\sqrt{3}/2\), which is negative. Hence, this angle’s terminal point is either in the second quadrant or the third quadrant. We know that in the first quadrant, \(\cos(\pi/6)=\sqrt{3}/2\). So, the reference angles in the second and third quadrant are \(\pi - \pi/6 = 5\pi/6\) and \(\pi + \pi/6 = 7\pi/6\), respectively.

Step 3: Verify the solution

Substitute these angles into the original equation to verify that \(\cos(5\pi/6)=-\sqrt{3}/2\) and \(\cos(7\pi/6)=-\sqrt{3}/2\). Thus, these are the correct solutions.

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry, providing a visual representation of angles and their corresponding sine and cosine values.

It is called a "unit" circle because it has a radius of 1. This makes calculations using it straightforward, as any point on the unit circle can be represented as \( (\cos x, \sin x) \), where \( x \) is the angle in radians.

In a unit circle:

• The x-coordinate of any point is the cosine of the angle.
• The y-coordinate is the sine of the angle.

The simplicity of the unit circle stems from it being a circle of radius one, focusing only on the essential trigonometric values. It can help us see why certain functions are positive or negative in different quadrants.

Cosine Function

The cosine function is an essential trigonometric function that relates to the x-coordinate of a point on the unit circle. It is denoted as \( \cos(x) \).

The function describes how the x-coordinate changes as an angle \( x \) increases or decreases. The cosine function is periodic, meaning it repeats values at regular intervals. Its period is \ ( 2\pi \, which means it repeats its cycle every \( 2\pi \) radians.

Here are key properties of the cosine function:

• It is positive in the first and fourth quadrants.
• Negative in the second and third quadrants.
• The range of values it can take is between -1 and 1.

Understanding these properties aids in solving equations like \( \cos(x) = -\sqrt{3}/2 \) by narrowing down the possible angle locations on the unit circle.

Reference Angles

Reference angles are a helpful concept when working with trigonometric functions like cosine. A reference angle is the smallest angle that an angle makes with the x-axis. It always falls between 0 and \( \pi/2 \) radians.

To find the reference angle for a given angle:

• If the angle is in the second quadrant, calculate \( \pi - \text{angle} \).
• If it is in the third quadrant, use \( \text{angle} - \pi \).

Reference angles simplify the problem by allowing us to determine values of trigonometric functions without knowing the exact position on the unit circle. For example, knowing \( \cos(\pi/6) = \sqrt{3}/2 \) helps us find that the related angles in the second and third quadrants, such as \( 5\pi/6 \) and \( 7\pi/6 \), share the same cosine value but with opposite signs.

Second Quadrant

The second quadrant of the unit circle lies between \( \pi/2 \) and \( \pi \) radians. In this region:

• Sine is positive, because the y-coordinates are positive.
• Cosine is negative, as the x-coordinates are negative.

Angles in the second quadrant are often referred to by subtracting the reference angle from \( \pi \), such as \( \pi - \text{reference angle} \). This method reveals that \( \cos(5\pi/6) = -\sqrt{3}/2 \), since it mirrors \( \cos(\pi/6) \) in terms of magnitude, but in the opposite direction when considering the x-axis.

Third Quadrant

Located between \( \pi \) and \( 3\pi/2 \), the third quadrant of the unit circle has its own trigonometric properties:

• Both sine and cosine are negative because the coordinates (x,y) here are both negative.

To find angles in the third quadrant, you add the reference angle to \( \pi \), resulting in expressions like \( \pi + \text{reference angle} \). This calculation shows how \( \cos(7\pi/6) = -\sqrt{3}/2 \), further using the negative x-coordinate to mirror positive cosine values of angles like \( \pi/6 \) found in the first quadrant.