Solving Cos Theta = Sqrt(3)/2

When we encounter a mathematical problem asking us to find $\theta$ if $\cos \theta = \frac{\sqrt{3}}{2}$, we're diving into the world of trigonometry, specifically looking for angles whose cosine value is a familiar one. The cosine function, in a nutshell, relates an angle in a right-angled triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. Knowing these fundamental relationships is key to solving this type of problem efficiently. Our goal is to identify which angle, or angles, when plugged into the cosine function, yield the specific value of $\frac{\sqrt{3}}{2}$. This value is a cornerstone in many trigonometric identities and calculations, making its associated angles particularly important to recognize. We'll explore the unit circle and special triangles to pinpoint the correct answer among the given options.

Understanding the Cosine Function and Special Angles

Let's delve deeper into why $\cos \theta = \frac{\sqrt{3}}{2}$ is such a significant equation in trigonometry. The value $\frac{\sqrt{3}}{2}$ is not arbitrary; it's one of the fundamental values associated with special angles. These special angles, typically 0, $\frac{\pi}{6}$ (30 degrees), $\frac{\pi}{4}$ (45 degrees), $\frac{\pi}{3}$ (60 degrees), and $\frac{\pi}{2}$ (90 degrees), and their multiples, have cosine and sine values that can be expressed using simple radicals and integers. They are derived from considering equilateral and isosceles right triangles. For instance, if you take an equilateral triangle with side length 2 and bisect one of its angles, you form two 30-60-90 right triangles. In such a triangle, the sides are in the ratio $1:\sqrt{3}:2$. The angle measuring 30 degrees ($\frac{\pi}{6}$) has a cosine of $\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$, and the angle measuring 60 degrees ($\frac{\pi}{3}$) has a cosine of $\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}$. Conversely, if you consider a 45-45-90 triangle (an isosceles right triangle), the sides are in the ratio $1:1:\sqrt{2}$. The cosine of 45 degrees ($\frac{\pi}{4}$) is $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$. Understanding these specific ratios and their corresponding angles is crucial for quickly solving problems like this one. The unit circle provides a visual representation where the cosine of an angle is the x-coordinate. As the angle increases from 0, the x-coordinate decreases from 1. At $\theta = \frac{\pi}{6}$, the x-coordinate is indeed $\frac{\sqrt{3}}{2}$.

Analyzing the Options

We are given four options for the value of $\theta$: A. $\frac{\pi}{6}$, B. $\frac{\pi}{3}$, C. $\frac{5\pi}{6}$, and D. $\frac{2\pi}{3}$. To find the correct answer, we need to evaluate the cosine of each of these angles and see which one equals $\frac{\sqrt{3}}{2}$.

• Option A: $\theta = \frac{\pi}{6}$ As discussed earlier, $\frac{\pi}{6}$ radians is equivalent to 30 degrees. In a 30-60-90 triangle, the side adjacent to the 30-degree angle is $\sqrt{3}$ and the hypotenuse is 2. Therefore, $\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$. This option is a strong candidate.

• Option B: $\theta = \frac{\pi}{3}$ This angle is 60 degrees. In the same 30-60-90 triangle, the side adjacent to the 60-degree angle is 1 and the hypotenuse is 2. So, $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$. This does not match our required value.

• Option C: $\theta = \frac{5\pi}{6}$ This angle is in the second quadrant (since $\frac{5\pi}{6}$ is between $\frac{\pi}{2}$ and $\pi$). The reference angle for $\frac{5\pi}{6}$ is $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$. Cosine is negative in the second quadrant. Therefore, $\cos \left(\frac{5\pi}{6}\right) = -\cos \left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$. This is not our target value.

• Option D: $\theta = \frac{2\pi}{3}$ This angle is also in the second quadrant (since $\frac{2\pi}{3}$ is between $\frac{\pi}{2}$ and $\pi$). The reference angle for $\frac{2\pi}{3}$ is $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$. Cosine is negative in the second quadrant. Therefore, $\cos \left(\frac{2\pi}{3}\right) = -\cos \left(\frac{\pi}{3}\right) = -\frac{1}{2}$. This is not our target value.

The Solution

Based on the analysis of each option, we found that only $\theta = \frac{\pi}{6}$ results in $\cos \theta = \frac{\sqrt{3}}{2}$. It's important to remember that trigonometric equations can have multiple solutions. For $\cos \theta = \frac{\sqrt{3}}{2}$, the general solution is $\theta = \pm \frac{\pi}{6} + 2n\pi$, where $n$ is an integer. This means angles like $\frac{\pi}{6}$, $\frac{11\pi}{6}$ (which is $-\frac{\pi}{6} + 2\pi$), and so on, all satisfy the equation. However, the question provides specific options, and we are looking for the one that is listed and correct. In the context of standard angles, $\frac{\pi}{6}$ is the principal value, and it is one of the options provided. Therefore, the correct answer is A. $\frac{\pi}{6}$. This problem highlights the importance of memorizing or being able to derive the trigonometric values for special angles and understanding how to determine the sign of trigonometric functions in different quadrants of the unit circle.

For further exploration into trigonometry and the unit circle, you can visit Khan Academy's Trigonometry section. This resource offers comprehensive explanations, practice exercises, and video tutorials that can deepen your understanding of these concepts.