Question

Determine the mistake that Sanesh made in the following work. Then, complete a correct solution.Solve \(2 \cos ^{2} x=\sqrt{3} \cos x .\) Express your answer(s) in degrees. Solution:$$\begin{aligned}\frac{1}{\cos x}\left(2 \cos ^{2} x\right) &=(\sqrt{3} \cos x) \frac{1}{\cos x} \\\2 \cos x &=\sqrt{3} \\\\\cos x &=\frac{\sqrt{3}}{2} \\ x &=30^{\circ}+360 \mathrm{n} \text { and } x=330^{\circ}+360^{\circ} \mathrm{n} \end{aligned}$$

Short answer

Sanesh missed \( x = 90^{\circ} + 360^{\circ} n \). The solutions are \( x = 90^{\circ} + 360^{\circ} n \), \( x = 30^{\circ} + 360^{\circ} n \), and \( x = 330^{\circ} + 360^{\circ} n \).

Step-by-step solution

Identify and isolate the cosine function

Start by recognizing that you need to simplify the equation. Given: \[ 2 \, \cos^2 x = \sqrt{3} \, \cos x \]To isolate the cosine function, move all terms to one side of the equation:\[ 2 \, \cos^2 x - \sqrt{3} \, \cos x = 0 \]

Factor out the common term

Factor out the common term \( \cos x \):\[ \cos x (2 \, \cos x - \sqrt{3}) = 0 \]

Solve for each factor separately

Set each factor equal to zero and solve:\[ \cos x = 0 \]and\[ 2 \, \cos x - \sqrt{3} = 0 \]

Find the values for \( \cos x = 0 \)

Solving \( \cos x = 0 \) gives:\[ x = 90^{\circ} + 360^{\circ} n \]

Solve for \( 2 \, \cos x - \sqrt{3} = 0 \)

Isolate \( \cos x \):\[ 2 \, \cos x = \sqrt{3} \]\[ \cos x = \frac{\sqrt{3}}{2} \]

Find the angle \( x \) for \( \cos x = \frac{\sqrt{3}}{2} \)

Identify the angles where \( \cos x = \frac{\sqrt{3}}{2} \):\[ x = 30^{\circ} + 360^{\circ} n \text{ and } x = 330^{\circ} + 360^{\circ} n \]

Combine all solutions

Combine all values for \( x \):\[ x = 90^{\circ} + 360^{\circ} n, \; x = 30^{\circ} + 360^{\circ} n, \; x = 330^{\circ} + 360^{\circ} n \]

Key concepts

Trigonometric Functions

Trigonometric functions play a crucial role in defining the relationships between the angles and sides of a triangle. The primary trigonometric functions are sine, cosine, and tangent, often denoted as sin, cos, and tan, respectively. Each function represents a ratio of two sides of a right-angled triangle. For instance, the cosine function (cos) represents the ratio of the adjacent side to the hypotenuse. Understanding these fundamental trigonometric functions is key to solving equations involving them, such as in the given exercise: \[2 \, \text{cos}^2 x = \sqrt{3} \, \text{cos} x\]This equation demonstrates the relationship involving the cosine function.

Quadratic Equations in Trigonometry

Quadratic equations can appear in trigonometric contexts, such as the provided problem: \[2 \, \text{cos}^2 x - \sqrt{3} \, \text{cos} x = 0\]This equation is quadratic because it involves the cos function squared. Solving these equations typically involves factoring, or using the quadratic formula. In our problem, we factor out the common term: \[\text{cos} x (2 \, \text{cos} x - \sqrt{3}) = 0\]Then, we solve for each factor separately. This method breaks down complex equations into simpler, more manageable parts, helping you to identify all possible solutions.

Degrees in Trigonometry

Degrees are a unit of measurement for angles. One full circle is 360 degrees. Understanding the degree system is vital when working with trigonometric functions, as it often provides the solutions in the context of a 360-degree circle. For instance, in the exercise solution: \[ \text{x} = 30^{\text{o}} + 360^{\text{o}} n \ \text{and} \text{x} = 330^{\text{o}} + 360^{\text{o}} n \ \text{x} = 90^{\text{o}} + 360^{\text{o}} n\] Here, 'n' represents any integer, showing that solutions can repeat every 360 degrees due to the periodic nature of trigonometric functions. This understanding helps in solving equations and interpreting their multiple solutions.

Cosine Function

The cosine function is one of the primary trigonometric functions. It represents the ratio of the adjacent side to the hypotenuse in a right triangle: \[ \text{cos} \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \ \]The cosine function also has several key properties: \

• It varies periodically, with a period of 360 degrees (or 2π radians).
• Its values range from -1 to 1.
• The function is symmetric about the y-axis (even function).

In solving our exercise: \[ \text{cos} x = 0 \ \text{or} \ 2 \, \text{cos} x = \sqrt{3} \]We use the cosine function's key properties and unit circle values to find all possible solutions. For instance, \text{cos} x = \frac{\sqrt{3}}{2} corresponds to angles 30 degrees and 330 degrees. Awareness of these values is instrumental in finding the correct and complete solution.