Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Solve. (a) \(\cos ^{2} x-\cos 2 x=0 \quad x \in(-\infty, \infty)\) (b) \(\sin x \cos x=\sqrt{3} \quad x \in[0,2 \pi]\) (c) \(\sin ^{2} x-\cos 2 x=0 \quad x \in[0,2 \pi]\) (d) \(-\cos ^{2} x+\frac{1}{2} \sin x+1=0 \quad x \in[0,2 \pi]\)

Step-by-Step Solution

Answer

\(x = \frac{\pi}{4}k (k ∈ Z)\), no solutions, \(x = \frac{\pi}{3}k (k ∈ [0,2\pi])\), \(x=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)

Step 1: Solving part (a)

The equation is \(\cos ^{2} x-\cos 2 x=0\). This can be simplified since \(\cos 2x = 2\cos^2x - 1\). Therefore, the equation becomes \(\cos^2x - 2\cos^2x + 1 = 0\), that simplifies to \(cos^2x = \frac{1}{2}\). This yields \(\cos x = \frac{\sqrt{2}}{2}\) and \(\cos x = -\frac{\sqrt{2}}{2}\) yielding the solutions \(x = \frac{\pi}{4}k\) where \(k\) is an integer.

Step 2: Solving part (b)

The equation is \(\sin x \cos x=\sqrt{3}\). Using the double angle formula, we can rewrite as \(\frac{1}{2}\sin 2x = \sqrt{3}\), then \(\sin 2x = 2\sqrt{3}\). Since \(\sin(2x)\) cannot be more than 1, there are no solutions in the given range.

Step 3: Solving part (c)

The equation is \(\sin ^{2} x-\cos 2 x=0\). This can be simplified as \(\sin ^{2} x-1+2\sin^2x =0\), which is \(2\sin^2x - 1 +\sin ^{2} x =0\), yielding the solutions \(x = \frac{\pi}{3}k\) where \(k\) is an integer in \([0,2\pi]\).

Step 4: Solving part (d)

The equation is \(-\cos ^{2} x+\frac{1}{2} \sin x+1=0 \). This can be simplified into \(-1 + \sin^2x + \frac{1}{2} \sin x +1 = 0\), which simplifies to \(\sin^2x + \frac{1}{2} \sin x = 0\). Factoring, we get \(\sin x(-2\sin x + 1)=0\), yielding the solutions \(x=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. They help in simplifying and solving various trigonometric equations. Some common trigonometric identities include the Pythagorean identity, like

• \[ \sin^{2}x + \cos^{2}x = 1 \]
• Double angle formulas, such as \( \cos(2x) = 2\cos^{2}x - 1 \)

In the exercise, identifying identities like \( \cos 2x = 2\cos^2x - 1 \) simplifies the original equation \( \cos^{2} x - \cos 2 x = 0 \), making it easier to find solutions. Recognizing these identities allows you to reorganize equations, making them much more manageable for further analysis and solving.

Double Angle Formulas

Double angle formulas express trigonometric functions of double angles (like \(2x\)) in terms of single angle functions. For instance, you have the double angle formula for cosine: \(\cos(2x) = 2\cos^2x - 1\). These formulas are especially useful in transforming complex trigonometric expressions into simpler ones.

In exercise (a), the identity \(\cos(2x)\) helps to break down \(\cos^{2} x - \cos 2 x = 0\) into a more straightforward equation, allowing easier solution steps. In exercise (b), recognizing the double angle formula for sine, \(\sin(2x) = 2\sin x \cos x\), simplifies solving by showing that \(\sin x \cos x\) is connected to the sine of double angles. Although there are no solutions due to the function range problem, the formula reveals insight into the problem's structure.

Unit Circle

The unit circle is a fundamental concept that helps visualize key trigonometric values and angles. It is a circle with a radius of one, centered at the origin of the coordinate plane. By referring to the unit circle, one can easily determine the sine and cosine values of standard angles. For instance:

• \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \)
• \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \)

The unit circle also helps in finding various angle solutions that correspond to specific sine and cosine values. For example, exercise (a) involves solving \( \cos x = \pm \frac{\sqrt{2}}{2} \), where students can use their knowledge of the unit circle to find multiple angle solutions by noting positions on the circle where cosine equals those values.

Solving Equations

Solving trigonometric equations involves a few strategic steps: identifying identities, simplifying, and factoring, which may lead to the use of substitution or other techniques. Initially, transform complex equations into simpler expressions using identities. Then find all solutions by considering both the principal value and all periodic solutions.

This process is evident in exercise (a), where using the identity \( \cos 2x = 2\cos^2x - 1 \) allowed the original equation to reduce to \( \cos^2 x = \frac{1}{2} \). This new equation gives \( \cos x = \pm \frac{\sqrt{2}}{2} \), leading to solutions derived from the unit circle. It's crucial to remember that the domain affects possible solutions, especially in bounded intervals like \([0, 2\pi]\).