Problem 39 - Precalculus with Limits

Question

In Exercises 39-44, solve the multiple-angle equation. \( \cos 2x = \dfrac{1}{2} \)

Step-by-Step Solution

Answer

The solution to the equation \( \cos 2x = \frac{1}{2} \) are \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \).

Step 1: Understand the equation

Firstly, it's needed to understand what the equation is telling. It's stated that the cosine of an angle is equal to \( \dfrac{1}{2} \). This provides the hint to think about the angles at which cosine equals \( \dfrac{1}{2} \).

Step 2: Find primary angles where cosine equals to 1/2

Next, recall the angles at which cosine equals \( \dfrac{1}{2} \). Using the unit circle, it's found that the angle that gives the cosine of \( \dfrac{1}{2} \) in the first quadrant is \( 60^{\circ} \) or \( \frac{\pi}{3} \) in radians. For the second quadrant, the angle is \( 300^{\circ} \) or \( \frac{5\pi}{3} \) in radians.

Step 3: Set the angle to equal to 2x

Now, instead of considering the original angles, set the multiple angle \( 2x \) equal to these valuues. This leads to the equations \( 2x= \frac{\pi}{3} \) and \( 2x= \frac{5\pi}{3} \).

Step 4: Solve for x

Finally, solve these equations for x. Divide both sides of each equation by 2. That gives \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \), which are the solutions to the given equation.

Key Concepts

Multiple Angle Identities

When dealing with trigonometric equations, multiple angle identities are useful tools. These identities help transform expressions involving multiple angles into simpler forms.
For example, the double angle identity for cosine is:

• \( \cos 2x = \cos^2 x - \sin^2 x \)
• This can also be rewritten using cosine and sine separately as:\( \cos 2x = 2\cos^2 x - 1 \)
• Or as another version: \( \cos 2x = 1 - 2\sin^2 x \)

Understanding these identities helps in simplifying or altering the form of a trigonometric equation. In the problem above, knowing that \( \cos 2x = \dfrac{1}{2} \) directly connects with primary angles is critical.
By setting the equation as a known cosine value, you can find your angle solutions efficiently.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is an essential tool in trigonometry for understanding angles and their corresponding sine and cosine values.
When it comes to finding the values of trigonometric functions at specific angles, the unit circle is invaluable. For cosine, you focus on the x-coordinate on the circle.

• At an angle of \( 0^{\circ} \), the cosine value is \( 1 \).
• At \( 60^{\circ} \) or \( \frac{\pi}{3} \), the cosine is \( \frac{1}{2} \). This is the angle found in the first quadrant of the circle.
• At \( 300^{\circ} \) or \( \frac{5\pi}{3} \), the cosine is also \( \frac{1}{2} \), found in the fourth quadrant.

Navigating the unit circle allows you to find these and other angles where cosine (or any trigonometric function) takes on specific values, making it easier to solve equations like the one given.

Cosine Function

The cosine function is a fundamental trigonometric function that associates an angle with its horizontal coordinate point on the unit circle. Understanding the pattern and properties of the cosine function is crucial.

• The cosine function is periodic, with a period of \( 2\pi \). This means it repeats its values every \( 2\pi \) radians.
• Notably, cosine is even. A function is called even when \( \cos x = \cos (-x) \).
• It ranges from -1 to 1 as it creates its distinctive wave pattern.
• Cosine is positive in the first and fourth quadrants and negative in the second and third.

This foundational knowledge assists in predicting cosine values at various points on the unit circle.
For example, knowing \( \cos 2x = \frac{1}{2} \) indicates a reoccurrence of such values, aiding in understanding the solution process for angle equations as in the given example.