Problem 36 - Algebra and Trigonometry

Question

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \cot \frac{3 \theta}{2}=-\sqrt{3} $$

Step-by-Step Solution

Answer

Solutions are \(\theta = \frac{4k\pi}{3}\), \(\theta = \frac{2\pi}{3} + \frac{4k\pi}{3}\), \(\theta = \frac{10\pi}{9} + \frac{4k\pi}{3}\), and \(\theta = \frac{4\pi}{3} + \frac{4k\pi}{3}\) for some integer \(k\), and where the value of \(\theta\) is in the interval [0, 2\(\pi\)).

Step 1: Identify the corresponding angle

First, identify the positive angle whose cotangent is \(\sqrt{3}\). Using the trigonometric ratios, you can determine that \(\cot \frac{\pi}{3} = \sqrt{3}\). So, \(\cot \alpha = \sqrt{3}\) where \(\alpha = \frac{\pi}{3}\).

Step 2: Find the cotangent's negative values

The cotangent is negative in the second and fourth quadrants. Therefore, the negative solutions for cotangent are \(\alpha_1 = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\) and \(\alpha_2 = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\) where \(0 \leq \alpha_1 < 2\pi\). The set of all solutions is \(\frac{3\theta}{2} = \alpha_1 + 2k\pi\) and \(\frac{3\theta}{2} = \alpha_2 + 2l\pi\) where \(k\) and \(l\) are integers.

Step 3: Solve for all possible values of \(\theta\)

Now solving for \(\theta\) we get the two following set of solutions: \(\theta = \frac{2\alpha_1}{3} + \frac{4k\pi}{3}\) and \(\theta = \frac{2\alpha_2}{3} + \frac{4l\pi}{3}\). Substituting the values of \(\alpha_1\) and \(\alpha_2\) and simplifying, we get the set of solutions on the interval [0, 2\(\pi\)) is \(\theta = \frac{4k\pi}{3}\), \(\frac{2\pi}{3} + \frac{4k\pi}{3}\), \(\frac{10\pi}{9} + \frac{4k\pi}{3}\) and \(\frac{4\pi}{3} + \frac{4k\pi}{3}\) where \(k\) is an integer and \(0 \leq \theta < 2\pi\).

Key Concepts

Understanding the Cotangent Function

The cotangent of an angle in trigonometry is a fundamental concept that often gets overlooked. It's the reciprocal of the tangent function. In mathematical terms, the cotangent of an angle \( \theta \) is defined as \( \cot \theta = \frac{1}{\tan \theta} \) or equivalently \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
This means the cotangent tells us the ratio between the adjacent side and the opposite side of a right-angled triangle when considering the given angle in that triangle.

• When solving trigonometric equations with cotangent, understanding its relationship with other ratios like sine and cosine is crucial.
• The value of \( \cot \theta \) can be positive or negative, depending on the quadrant in which the angle \( \theta \) is situated.

It is important to remember that cotangent, like other trigonometric functions, has specific values at notable angles like \( \pi/3 \), \( \pi/4 \), etc., commonly used in trigonometric equations.

Finding Angle Solutions

In trigonometric equations, finding angle solutions involves determining all possible angles that satisfy the equation within a given range—in this case, \([0, 2\pi)\).
When you see an equation like \( \cot \frac{3\theta}{2} = -\sqrt{3} \), finding solutions depends heavily on understanding where the cotangent function takes specific values.

• First, we need to identify reference angles from known cotangent values. For example, \( \cot \frac{\pi}{3} = \sqrt{3} \).
• Next, we assess in which quadrants the cotangent is negative, since the original problem involves a negative cotangent value.

For \( \cot \alpha = \sqrt{3} \), the corresponding negative solutions are found in the second and fourth quadrants, giving us angles \( \alpha_1 = \frac{2\pi}{3} \) and \( \alpha_2 = \frac{5\pi}{3} \). These steps help in building the set of possible angle solutions.

Navigating Multiple Angle Equations

When dealing with multiple angle equations like \( \cot \frac{3\theta}{2} = -\sqrt{3} \), the process involves an extra step of considering the multiple factor before solving for \( \theta \).

• The given equation is of the form \( \cot(n\theta) = a \). Here, \( n = \frac{3}{2} \).
• Once you find the reference angles (\( \alpha_1 \) and \( \alpha_2 \)), you need to adjust for the angle's multiplication by solving \( \frac{3\theta}{2} = \alpha + 2k\pi \).

Finally, solving for \( \theta \) involves rearranging to isolate \( \theta \). This adjustment means adding a whole cycle of solutions, manipulated by the multiplying factor, to account for the angles' complete rotation through the possible values.

Exploring Trigonometric Ratios

Trigonometric ratios are the backbone of solving equations like \( \cot \frac{3\theta}{2} = -\sqrt{3} \). These ratios—sine, cosine, tangent, cotangent, secant, and cosecant—define the relationship between the sides and angles of triangles.

• Each ratio has specific values for standard angles such as \( 0, \pi/6, \pi/4, \pi/3, \pi/2, \) that allow us to deduce the solutions to trigonometric equations.
• Understanding trigonometric identities and reciprocal identities (e.g., \( \cot \theta = \frac{1}{\tan \theta} \)) enriches our ability to manipulate and solve equations.

These ratios not only help find angles but also facilitate interpretations of periodic and oscillatory phenomena in various fields of science and engineering. Recognizing the significance of these basic identities paves the way for tackling more complex equations.