Problem 23 - 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. $$\sqrt{2} \sin 3 x-1=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{12} + \frac{k\pi}{3}\) in radians and \(x = 15° + 60k°\) in degrees.

Step 1: Isolate the Trigonometric Function

The given equation is \(\sqrt{2} \sin (3x) - 1 = 0\). To isolate the sine function, we need to add 1 to both sides: \[\sqrt{2} \sin (3x) = 1\]. Then, divide both sides by \(\sqrt{2}\): \[\sin (3x) = \frac{1}{\sqrt{2}}\].

Step 2: Determine the General Solution for the Sine Function

We know that \(\sin \theta = \frac{1}{\sqrt{2}}\) corresponds to \(\theta \) being \(\frac{\pi}{4} + k\pi\) for any integer \(k\), since sine has a period of \(\pi\) and is positive in the first and second quadrants.

Step 3: Solve for \(3x\) Using General Solution

Using the general solution, \(3x = \frac{\pi}{4} + k\pi\) for integer \(k\). This equation accounts for all possible angle measures in radians that solve the sine equation.

Step 4: Solve for \(x\) by Dividing Both Sides by 3

Divide the equation \(3x = \frac{\pi}{4} + k\pi\) by 3 to solve for \(x\). Thus, we have \[x = \frac{\pi}{12} + \frac{k\pi}{3}\].

Step 5: Convert Possible Angle Measures into Degrees

Since the problem asks for solutions in both radians and degrees, let's convert \(x = \frac{\pi}{12} + \frac{k\pi}{3}\) into degrees: \(\frac{\pi}{12}\) is equivalent to 15°, and \(\frac{k\pi}{3}\) is equivalent to 60° * \(k\). Therefore, the solution in degrees is \(x = 15° + 60k°\).

Key Concepts

Radian Measure

Radian measure is a way of measuring angles using the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians express the angle through the length of the arc. The formula to convert degrees to radians is:

• Multiply the number of degrees by \( \frac{\pi}{180} \).

For example, a 180° angle is equivalent to \( \pi \) radians. This is because the circumference of the circle is \( 2\pi \) and a semicircle is half of that length.
It's a more natural unit in mathematics, especially when dealing with trigonometric functions because it relates the angle's size directly to the arc it subtends.
Utilizing radian measure in solving trigonometric equations allows for more direct calculations and is essential when angles are involved in calculus and other advanced fields.

Degree Measure

Degree measure is the most common way we learn to measure angles. It divides a full circle into 360 equal parts. This means each degree is \(\frac{1}{360}\) of a circle.
Degrees are easier to visualize for most practical purposes because they are widely used in everyday contexts like navigation or carpentry.

• To convert an angle from radians to degrees, use the formula \( \theta(\text{degrees}) = \theta(\text{radians}) \times \frac{180}{\pi} \).

For instance, an angle of \( \pi/4 \) radians converts to 45°.
In the context of solving trigonometric equations, having solutions in both degrees and radians can be helpful because it allows us to understand the angle in both mathematical and practical terms.

Sine Function

The sine function is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.
Mathematically, it is expressed as:

• \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

The sine function is periodic with a period of \(2\pi \) radians, meaning it repeats its values every \(2\pi \) radians.
It is useful in many aspects of science and engineering because it describes wave patterns, such as sound waves or light waves.
The sine function is also central in solving trigonometric equations like the one we explored. For example, solving \(\sin (3x) = \frac{1}{\sqrt{2}}\) relies on knowing specific sine values like \(\frac{\pi}{4} \), where sine equals this ratio.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals. The sine function is one of the most well-known periodic functions.

• For the sine function, this interval is its period \(2\pi \) radians or 360°.

This property makes it very predictable and applicable in modeling situations that repeat or cycle, like seasonal weather patterns or electrical currents.
When solving trigonometric equations, understanding periodicity allows us to find all potential solutions.
For instance, \(\sin(x) = \frac{1}{\sqrt{2}}\) yields multiple solution angles because of the sine wave's repetitive nature. By recognizing that occurrences repeat every period, we can write general solutions that account for all possible angles, using expressions like \(x = \frac{\pi}{12} + \frac{k\pi}{3} \). This formula shows how periodicity generates a cycle of solutions.