Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$-2 \cos 2 x=\sqrt{3}$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5\pi}{12} + k\pi\), \(x = \frac{7\pi}{12} + k\pi\) in radians and 75°, 105° plus 180°k in degrees.

Step 1: Isolate the Cosine Function

We are given the equation \(-2 \cos 2x = \sqrt{3}\). First, divide both sides by \(-2\) to isolate the cosine function:\[ \cos 2x = -\frac{\sqrt{3}}{2} \]

Step 2: Determine Reference Angles in Radians

Now, identify where \(\cos(\theta) = -\frac{\sqrt{3}}{2}\) on the unit circle. This cosine value corresponds to angles where cosine is negative, specifically in the second and third quadrants where the reference angle \(\frac{\pi}{6}\) applies.

Step 3: Calculate General Solutions for \(2x\) in Radians

From the unit circle, the angles of interest where \(\cos 2x = -\frac{\sqrt{3}}{2}\) are \(2x = \pi - \frac{\pi}{6}\) and \(2x = \pi + \frac{\pi}{6}\), which are:\[ 2x = \frac{5\pi}{6} + 2k\pi \]\[ 2x = \frac{7\pi}{6} + 2k\pi \] where \(k\) is an integer.

Step 4: Solve for \(x\) in Radians

Divide both sides of the general solutions by 2 to solve for \(x\):\[ x = \frac{5\pi}{12} + k\pi \]\[ x = \frac{7\pi}{12} + k\pi \]

Step 5: Convert Radian Answers to Degrees

To convert radians to degrees, use the conversion factor \(180^\circ/\pi\):\[ x = \frac{5\pi}{12} = 75^\circ \quad \text{and} \quad x = \frac{5\pi}{12} + 180^\circ k \]\[ x = \frac{7\pi}{12} = 105^\circ \quad \text{and} \quad x = \frac{7\pi}{12} + 180^\circ k \]

Step 6: Determine the Least Nonnegative Angles

Review the angles derived (75 and 105 degrees) and note they are already nonnegative and the smallest possible solutions for \(x\) considering the periodic nature of cosine.

Key Concepts

Radians

Radians are an essential measure of angles used primarily in trigonometry and calculus. Instead of measuring angles in degrees, radians offer a mathematical advantage in calculations and analysis. A radian is defined as the angle subtended by an arc of a circle that is equal in length to the radius of the circle. This means that:

• 1 complete revolution (360 degrees) is equal to \(2\pi\) radians.
• 1 radian is approximately equivalent to 57.2958 degrees.

Knowing how to convert between radians and degrees is useful. For example, converting \(75^{\circ}\) to radians involves multiplying by \(\pi/180\), resulting in approximately \(1.3089\) radians.

Degrees

Degrees are a traditional measure of angles that are widely used in everyday contexts. It's simple to understand and visualize. Angles are measured in a full circle being 360 degrees.

• 90 degrees represents a right angle, forming a square corner.
• 180 degrees forms a straight line.
• 360 degrees forms a complete circle.

Degrees are sometimes preferred for practical purposes where the circle needs to be divided into equal parts, such as in navigation and horology. The conversion from radians to degrees is crucial and involves using the factor \(180/\pi\). For example, converting \(5\pi/12\) radians to degrees is calculated by \((5\pi/12) \times (180/\pi) = 75^{\circ}\).

Unit Circle

The unit circle is a powerful tool in trigonometry that helps visualize and solve angles and trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane. In the unit circle:

• Angles are measured from the positive x-axis, counterclockwise.
• The coordinates of any point on the circle are \((\cos\theta, \sin\theta)\).

This means that if you know the angle, you can directly find the cosine and sine values. Since cosine values appear as the x-coordinates, \( \cos \theta = -\sqrt{3}/2 \) can tell us the relevant angles are in the second and third quadrants, corresponding to specific reference angles.

Reference Angles

Reference angles are a core concept in understanding trigonometric solutions within different quadrants. A reference angle is the acute angle between the terminal side of the angle and the horizontal axis. This means:

• The reference angle is always between 0 and \(90^{\circ}\) or \(0\) and \(\pi/2\) radians.
• It helps simplify problems as any angle can be reduced to a reference angle.

To find the cosine or sine of an angle in any quadrant, first determine the reference angle and then adjust the sign based on the quadrant. In our problem, the reference angle for \(\cos \theta = -\sqrt{3}/2 \) is \( \pi/6 \), or \(30^{\circ}\), which tells us about positions in both the second and third quadrants.