Problem 32 - 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. $$\sin 2 x=2 \cos ^{2} x$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{4} + n\pi \) and \( x = \frac{\pi}{2} + n\pi \), where \(n\) is an integer.

Step 1: Rewrite the equation using a trigonometric identity

The given equation is \(\sin 2x = 2 \cos^2 x\). We know from a trigonometric identity that \(\sin 2x = 2 \sin x \cos x\). Replace \(\sin 2x\) in the equation: \[ 2 \sin x \cos x = 2 \cos^2 x \] We can divide both sides by 2: \[ \sin x \cos x = \cos^2 x \]

Step 2: Simplify the equation

Simplify the equation \(\sin x \cos x = \cos^2 x\) by dividing both sides by \(\cos x\) (assuming \(\cos x eq 0\)):\[ \sin x = \cos x \] This implies that the angle \(x\) is where the sine of \(x\) equals the cosine of \(x\).

Step 3: Solve for exact solutions

The equation \(\sin x = \cos x\) occurs where the tangent of \(x\) is 1. This occurs at angles of 45 degrees or \(\frac{\pi}{4}\) radians, and every \(180\) degrees or \(\pi\) radians afterwards: \[ x = \frac{\pi}{4} + n\pi \] where \(n\) is an integer, representing the periodicity of the tangent function.

Step 4: Consider conditions where \(\cos x = 0\)

Since we divided by \(\cos x\) in Step 2, we also need to consider when \(\cos x = 0\). This happens at odd multiples of \(\frac{\pi}{2}\):\[ x = \frac{\pi}{2} + n\pi \] where \(n\) is an integer.

Step 5: Combine solutions for the equation

From Steps 3 and 4, we have two sets of solutions: - \( x = \frac{\pi}{4} + n\pi \)- \( x = \frac{\pi}{2} + n\pi \).These solutions cover all angles where the original equation holds true.

Key Concepts

Trigonometric Identities

Trigonometric identities are useful relationships between trigonometric functions that help simplify and solve equations. In the given exercise, we transform the problem using the identity for double angles: \(\sin 2x = 2 \sin x \cos x\). This identity turns the original equation \(\sin 2x = 2 \cos^2 x\) into a new form: \(2 \sin x \cos x = 2 \cos^2 x\). Utilizing such identities is essential for simplifying expressions and solving equations efficiently. These identities also help uncover more insights about the trigonometric functions' nature and properties. Remember, becoming familiar with these identities can greatly assist in tackling various trigonometric problems. They often serve as key tools in transforming complex equations into more manageable forms.

Periodicity

Trigonometric functions exhibit periodic behavior, meaning they repeat their values over regular intervals. This property is crucial when finding all possible solutions to trigonometric equations. In this exercise, the periodicity of the tangent function plays a significant role. Since the tangent function repeats every \(\pi\) radians or \(180\) degrees, it helps us determine solutions that recur at intervals like \(x = \frac{\pi}{4} + n\pi\) where \(n\) is any integer. Additionally, the periodic nature of \(\cos x = 0\) leads to the solutions at \(x = \frac{\pi}{2} + n\pi\). Understanding periodicity allows us to extend single solutions to infinite series of solutions within the trigonometric function's cycle.

Exact Solutions

Exact solutions refer to solutions expressed in terms of exact angle measures without approximations. In trigonometry, these are typically expressed in fractions of \(\pi\) when dealing with radians or simple degree measures. For the equation \(\sin x = \cos x\), an exact solution is \(x = \frac{\pi}{4}\). Such solutions emerge from solving trigonometric equations where values correspond to well-known angles derived from the unit circle. Calculating exact solutions is beneficial as it gives precise results, representing fundamental angles without rounding errors. Grasping the concept of exact solutions helps in understanding fundamental relationships between trigonometric functions.

Angle Measures

Angle measures in trigonometry can be in radians or degrees. Being versatile with both systems is essential, especially when solving equations. In this exercise, we work with solutions in radians like \(\frac{\pi}{4}\) and \(\frac{\pi}{2}\), while also acknowledging equivalent degree measures like \(45^{\circ}\) and \(90^{\circ}\). Recognizing when to use radians versus degrees depends on the context of the problem. Radians, derived from the radius of a circle, are often favored in calculus and more advanced mathematics, offering simplicity in periodicity and mathematical analysis. On the other hand, degrees are intuitive and widely used in everyday contexts. Understanding and converting between these measures can enhance your mathematical efficiency and fluency.