Find the Exact Value arcos (-( square root of 3)/2)
The question is asking for the calculation of the inverse trigonometric function of arccosine (also denoted as cos^(-1)) for a specific value, which in this case is the negative square root of three divided by 2. Essentially, it is requesting the angle (in degrees or radians) whose cosine equals -√3/2. The task is to determine the exact angle that has this cosine value without using numerical approximation methods.
$arccos \left(\right. - \frac{\sqrt{3}}{2} \left.\right)$
Answer
Solution:
Step 1:
Express the given problem using mathematical notation: $arccos\left(-\frac{\sqrt{3}}{2}\right)$.
Step 2:
Identify the angle whose cosine is $-\frac{\sqrt{3}}{2}$. The angle is $\frac{5\pi}{6}$.
Step 3:
Present the solution in various formats. In its exact form, the solution is $\frac{5\pi}{6}$. In decimal form, it is approximately $2.61799387$.
Knowledge Notes:
To solve for the arccosine (inverse cosine) of a given value, we must understand the cosine function and its properties. The cosine function relates an angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse.
Inverse Trigonometric Functions: The arccosine function is the inverse of the cosine function. It takes a value and returns the angle whose cosine is that value. Since the range of arccosine is restricted to $[0, \pi]$, there is only one angle in this interval for each value.
Reference Angles: A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It is used to find the cosine of angles that are not in the first quadrant by relating them to an acute angle with the same cosine value.
Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system. It is a helpful tool for finding the cosine of an angle. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle formed by the point, the origin, and the positive x-axis.
Exact Values: Certain angles have exact values for their trigonometric functions, which are often derived from the unit circle or special triangles (like the 30-60-90 triangle). For example, the cosine of $60^\circ$ (or $\frac{\pi}{3}$) is $\frac{1}{2}$, and the cosine of $120^\circ$ (or $\frac{2\pi}{3}$) is $-\frac{1}{2}$ due to symmetry in the unit circle.
Radians and Degrees: Trigonometric functions can be expressed in radians or degrees. Radians are often used in higher mathematics due to their natural appearance in calculus and other areas. The angle $\frac{5\pi}{6}$ radians is equivalent to $150^\circ$.
In this problem, we are asked to find the arccosine of $-\frac{\sqrt{3}}{2}$. By recognizing that the cosine of $150^\circ$ (or $\frac{5\pi}{6}$ radians) is $-\frac{\sqrt{3}}{2}$, we can determine that $arccos\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6}$.
