Problem 54 - Algebra and Trigonometry

Question

(a) Find all solutions of the equation. (b) Use a calculator to solve the equation in the interval \([0,2 \pi),\) correct to five decimal places. $$2 \sin 2 x-\cos x=0$$

Step-by-Step Solution

Answer

Solutions: \( x \approx 0.25268, 1.57080, 2.88891, 4.71239 \) in \([0, 2\pi)\).

Step 1 - Apply a Trigonometric Identity

The given equation is \( 2 \sin 2x - \cos x = 0 \). Use the double angle identity \( \sin 2x = 2 \sin x \cos x \), which will allow you to express the equation in terms of \( \sin x \) and \( \cos x \). The equation becomes: \[ 4 \sin x \cos x - \cos x = 0 \].

Step 2 - Factor the Equation

Factor \( \cos x \) from the equation: \( \cos x (4 \sin x - 1) = 0 \). This gives two possible equations to solve: \( \cos x = 0 \) and \( 4 \sin x - 1 = 0 \).

Step 3 - Solve for \( \cos x = 0 \)

Find the values of \( x \) where \( \cos x = 0 \). This occurs when \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Step 4 - Solve for \( 4 \sin x - 1 = 0 \)

Rearrange the equation to get \( \sin x = \frac{1}{4} \). Then find \( x \) using the unit circle or a calculator. In the interval \([0, 2\pi)\), this yields \( x = \arcsin \left( \frac{1}{4} \right) \) and \( x = \pi - \arcsin \left( \frac{1}{4} \right) \).

Step 5 - Use Calculator to Get Exact Values

Using a calculator, find \( \arcsin \left( \frac{1}{4} \right) \). The approximate value is \( x \approx 0.25268 \). The second solution is \( x \approx \pi - 0.25268 = 2.88891 \).

Step 6 - Compile All Solutions in the Given Interval

The potential solutions for \( x \) from Step 3 in the interval \([0, 2\pi)\) are \( x = \frac{\pi}{2} \approx 1.57080 \) and \( x = \frac{3\pi}{2} \approx 4.71239 \). From Step 4, the additional solutions are \( x \approx 0.25268 \) and \( x \approx 2.88891 \).

Key Concepts

Double Angle Identity

A double angle identity is a trigonometric expression that involves functions of twice the angle, for example, \( 2x \). These identities help simplify complex trigonometric equations, making them easier to solve. In our exercise, we begin with \( 2\sin 2x - \cos x = 0 \).
Here, the double angle identity \( \sin 2x = 2\sin x\cos x \) is used to rewrite the equation. By substituting this identity, the equation becomes \( 4\sin x\cos x - \cos x = 0 \).
This transformation allows us to focus on the basic trigonometric functions, \( \sin x\) and \( \cos x\), which are more straightforward to work with. The application of identities like these is essential to breaking down equations into forms that we can easily manage and solve.
Recognizing and using these identities correctly is crucial for anyone studying trigonometry, especially when dealing with equations involving multiple angles.

Factoring Trigonometric Equations

Factoring is another key strategy in solving trigonometric equations. Once the equation \( 4\sin x\cos x - \cos x = 0 \) is written using the double angle identity, we see that factoring becomes a viable option.
In this situation, we can factor \( \cos x \) out of the equation, leading to \( \cos x (4 \sin x - 1) = 0 \).
This equation is now separated into two simpler equations: \( \cos x = 0 \) and \( 4\sin x - 1 = 0 \). By factoring, we essentially create smaller, more manageable problems to solve.

• For \( \cos x = 0 \), we find solutions where the cosine function equals zero.
• For \( 4\sin x - 1 = 0 \), we solve for \( \sin x = \frac{1}{4} \).

Factoring like this simplifies the original equation and lets us pinpoint specific values for \( x \) more easily. This step is often necessary when working with polynomials and trigonometric expressions, emphasizing the importance of factoring in mathematical problem-solving.

Unit Circle

The unit circle is a powerful tool in trigonometry that helps us understand angle measures and associated trigonometric values. It represents all the angle positions between 0 and \( 2\pi \).
For this exercise, we use the unit circle to determine the values of \( x \) when \( \cos x = 0 \). These occur at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), aligning with where the unit circle's cosine value is zero.
Additionally, the unit circle aids in visualizing solutions for equations like \( \sin x = \frac{1}{4} \). While the exact values are calculated using a calculator, the unit circle conceptually shows us that \( \sin x = 0.25 \) corresponds to specific points on the circle.

• Allows quick identification of angle measures.
• Provides visualization of trigonometric function values.

Overall, the unit circle not only offers insight into solving equations graphically but also reinforces the understanding of how angles and functions are interconnected.

Calculator Solutions for Trigonometry

When solutions for trigonometric equations demand precision, calculators become indispensable. Especially when finding exact values for functions like \( \arcsin \), which are not easily expressed using the unit circle alone.
In this exercise, we calculate \( x \) where \( \sin x = \frac{1}{4} \). Using a calculator, we find one solution as \( x \approx 0.25268 \). To find the second solution, calculate \( \pi - 0.25268 \approx 2.88891 \), utilizing the periodic nature of sine functions.
It’s vital to remember:

• Always set the calculator to the appropriate mode (radians for this exercise).
• Round the solutions to the desired decimal places as instructed.
• Check for possible additional solutions within specified intervals.

Using calculators simplifies complex computations, allowing you to double-check and explore solutions accurately, especially within specified intervals like \([0, 2\pi)\). This solidifies your foundational skills in trigonometry.