Problem 5 - Algebra and Trigonometry

Question

\(3-10=\) Find the exact value of each expression, if it is defined. $$ \begin{array}{llll}{\text { (a) } \cos ^{-1}(-1)} & {\text { (b) } \cos ^{-1} \frac{1}{2}} & {\text { (c) } \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \pi \), (b) \( \frac{\pi}{3} \), (c) \( \frac{5\pi}{6} \).

Step 1: Understanding the Inverse Cosine Function

The function \( \cos^{-1}(x) \) is the inverse of the cosine function, which means it returns the angle \( \theta \) whose cosine is \( x \). The range of \( \cos^{-1}(x) \) is \([0, \pi]\). We will evaluate each expression within this range to find the angles that correspond to the given values.

Step 2: Solving part (a): \( \cos^{-1}(-1) \)

To find \( \cos^{-1}(-1) \), we need to determine the angle \( \theta \) such that \( \cos(\theta) = -1 \). The angle whose cosine is -1 within the range \([0, \pi]\) is \( \theta = \pi \). Thus, \( \cos^{-1}(-1) = \pi \).

Step 3: Solving part (b): \( \cos^{-1}\left(\frac{1}{2}\right) \)

To find \( \cos^{-1}\left(\frac{1}{2}\right) \), we seek the angle \( \theta \) with \( \cos(\theta) = \frac{1}{2} \) within the range \([0, \pi]\). This angle is \( \theta = \frac{\pi}{3} \). Thus, \( \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \).

Step 4: Solving part (c): \( \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) \)

For \( \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) \), we need to find the angle \( \theta \) such that \( \cos(\theta) = -\frac{\sqrt{3}}{2} \) within the range \([0, \pi]\). This angle is \( \theta = \frac{5\pi}{6} \). Therefore, \( \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6} \).

Key Concepts

Inverse Cosine Function

The inverse cosine function, written as \( \cos^{-1}(x) \) or sometimes \( \text{arccos}(x) \), is a special function used to find the angle whose cosine value is \( x \). This means if you know the value of the cosine of an angle, you can determine the actual angle itself using the inverse cosine function. This function is only applicable when the cosine value is between -1 and 1, as those are the limits for any cosine value.
These angles are usually expressed in radians when using inverse trigonometric functions.

• Example: \( \cos^{-1}(0) \) will give you an angle of \( \frac{\pi}{2} \) rad because the cosine of \( \frac{\pi}{2} \) is 0.
• If you input a number outside the range of [-1,1], the function is not defined, as cosine values cannot reach beyond these limits.

By applying \( \cos^{-1} \), we are essentially reversing the cosine function to find which angle initially gave us the value of \( x \). This is particularly useful in solving various trigonometric equations.

Range of Inverse Trig Functions

The range of inverse trigonometric functions is crucial because it defines the possible output values for these functions. For the inverse cosine function, \( \cos^{-1}(x) \), the range is restricted to \([0, \pi]\) radians. This means the angle the function returns will always be between 0 and \( \pi \) radians.

• Why \([0, \pi]\)? The cosine function is symmetric about 0 in this range and covers all values of cosine from -1 to 1.
• Keep in mind: this range guarantees a unique solution for each input value \( x \).

This restriction is essential because without it, one cosine value would correspond to infinitely many angle measures around the unit circle. By setting this range, the inverse cosine function remains a true function (each input corresponds to exactly one output).
This principle applies similarly to other inverse trigonometric functions with their respective ranges.

Trigonometric Equations

Trigonometric equations involve unknown angles in expressions with trigonometric functions like sine, cosine, tangent, etc. Solutions typically revolve around applying inverse trigonometric functions to "undo" the trigonometric function part, effectively revealing the angle.For instance, solving \( \cos^{-1}(\frac{1}{2}) \) involves determining which angle results in a cosine of \( \frac{1}{2} \). Employing trigonometric identities and tables, we know this corresponds to an angle of \( \frac{\pi}{3} \). This is because the cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \).

• The fundamental objective is to express the angle alone on one side of the equation after simplifying the expression using inverse functions.
• The usage of inverse functions often simplifies complex trigonometric equations into manageable forms.

Understanding how to work with inverse trigonometric functions is essential in solving not only simple trigonometric equations but also more involved problems that may arise in calculus.