Problem 30 - 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. $$\cos \theta-1=\cos 2 \theta$$

Step-by-Step Solution

Answer

The solutions are \(\theta = 60^\circ, 90^\circ, 270^\circ, 300^\circ\) or \(\theta = \frac{\pi}{3}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{3}\).

Step 1: Substitute and Simplify

We start with the given equation \( \cos \theta - 1 = \cos 2\theta \). We can use the double angle identity for cosine, \( \cos 2\theta = 2\cos^2\theta - 1 \), to rewrite the equation as \( \cos \theta - 1 = 2\cos^2\theta - 1 \).

Step 2: Rearrange the Equation

Next, rearrange the equation: \( \cos \theta - 2\cos^2\theta = 0 \). Factoring out \( \cos \theta \), we get the expression \( \cos \theta (1 - 2\cos \theta) = 0 \).

Step 3: Solve the Factored Equation

Set each factor equal to zero: \( \cos \theta = 0 \) and \( 1 - 2\cos \theta = 0 \). For \( \cos \theta = 0 \), the solutions are \( \theta = 90^\circ, 270^\circ, 450^\circ, \ldots \) or in radians \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \). For \( 1 - 2\cos \theta = 0 \), simplifying gives \( \cos \theta = \frac{1}{2} \), so \( \theta = 60^\circ, 300^\circ \) or in radians \( \theta = \frac{\pi}{3}, \frac{5\pi}{3} \).

Step 4: List the Solutions

Combine all solutions and express them with the least nonnegative angle measures: In degrees, \( \theta = 60^\circ, 90^\circ, 270^\circ, 300^\circ \). In radians, \( \theta = \frac{\pi}{3}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{3} \).

Key Concepts

Radians

Radians are a unit of angular measure used in mathematics. One full rotation is equal to \( 2\pi \) radians. This unit is often preferred in trigonometry because it naturally incorporates the circle's radius and arc length. The formula relates the angle in radians to the arc length and radius: \( Angle = \frac{Arc \ Length}{Radius} \). Understanding radians is crucial when dealing with angular measurements in trigonometric equations.

In many cases, solving trigonometric equations leads to solutions expressed in radians, especially in calculus and advanced applications. When converting from degrees to radians, use the formula:

• \( Degree \ Value \times \frac{\pi}{180} \)

Conversely, to convert from radians to degrees, the formula is:

• \( Radian \ Value \times \frac{180}{\pi} \)

For example, \(\frac{\pi}{3}\) radians can be converted to degrees as follows: \(\frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ\).

Degrees

Degrees are the most commonly known unit for measuring angles. A full circle is 360 degrees. This system is familiar to most students because it is used in everyday applications, like with compasses and protractors.

In trigonometry, understanding how to navigate between degrees and radians is essential for solving problems, especially since many trigonometric functions and identities can be more easily expressed in radians. However, when calculations need to match real-world scenarios, degrees are often used.

Key angles frequently encountered in trigonometry include:

• 0°
• 90°
• 180°
• 270°
• 360°

Understanding these angles helps in graphing and solving equations like cosine equations, since these represent the axes and cardinal directions in a unit circle.

Cosine

Cosine is one of the primary trigonometric functions, commonly written as \(\cos \). It relates to the angle in a right triangle as the ratio of the adjacent side to the hypotenuse.

The cosine function is crucial for solving trigonometric equations and, in particular, is heavily utilized in transformations using the double angle formulas. The basic properties and values of cosine at key angles include:

• \(\cos(0°) = 1\)
• \(\cos(90°) = 0\)
• \(\cos(180°) = -1\)
• \(\cos(360°) = 1\)

Understanding the cosine function's behavior helps determine solutions to equations when factoring and finding zeros or intersections on a graph. Moreover, knowing that \(\cos\theta = 0\) can result in angles such as 90° and 270°, is critical for exact solutions.

Double Angle Formula

The Double Angle Formula is a pivotal identity that simplifies expressions involving trigonometric functions with double angles. For cosine, the formula is \(\cos(2\theta) = 2\cos^2(\theta) - 1\). This formula is instrumental in transforming complex trigonometric equations into solvable linear or quadratic equations.

It is widely used when simplifying expressions or verifying identities in trigonometry. By applying the double-angle formula, terms involving \(\cos(2\theta)\) can be rewritten in terms of \(\cos(\theta)\). This is crucial for solving equations like the one from the original exercise where transforming \(\cos 2\theta\) allows us to use simpler algebraic techniques.

In problems, use this formula to express double-angle terms as:

• \(\cos(2\theta) = 2\cos^2(\theta) - 1\)
• Alternatively in terms of sine: \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)

These identities not only simplify expressions but also help uncover solutions in equations where multiple angles appear.