Problem 12 - Thomas Calculus in SI Units

Question

Find the values. $$\cot \left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$$

Step-by-Step Solution

Answer

The value is \(-\frac{\sqrt{3}}{3}\).

Step 1: Understanding Inverse Sine

The expression given is \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \). This asks us to find an angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \). In the unit circle, the sine function is negative in the third and fourth quadrants.

Step 2: Determine the Principal Angle

The range for \( \sin^{-1}(x) \) is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). Therefore, \( \theta \) should be in the fourth quadrant. The angle corresponding to \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) is \( \theta = -\frac{\pi}{3} \).

Step 3: Find the Cotangent

Now we need to find \( \cot(-\frac{\pi}{3}) \). We know that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). For \( \theta = -\frac{\pi}{3} \), \( \tan(-\frac{\pi}{3}) = -\sqrt{3} \). Therefore, \( \cot(-\frac{\pi}{3}) = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \sin^{-1} \) (also called arcsin), are crucial for finding angles when given a trigonometric ratio. They essentially reverse the trigonometric functions. For instance, if \( \sin(\theta) = x \), then \( \sin^{-1}(x) = \theta \). These inverse functions have specific ranges to ensure each input delivers a unique output, which makes them proper functions.

• For \( \sin^{-1}(x) \), the range is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Inverse trigonometric functions help translate points on the unit circle back into angles, which can then be used for further trigonometric calculations like finding cotangent.

Cotangent

Cotangent, denoted by \( \cot(\theta) \), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. The relationship can be written as:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• \( \cot(\theta) \) can also be expressed as \( \frac{\cos(\theta)}{\sin(\theta)} \).

It is particularly useful when the calculation of tangent directly isn't feasible or straightforward. For angles where \( \tan(\theta) \) is known to be undefined, such as at multiples of \( \frac{\pi}{2} \), \( \cot(\theta) \) will also be undefined. When calculating \( \cot(\theta) \) and ensuring your equation is consistent, consider rationalizing the denominator for simplicity.

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is a foundational concept in trigonometry, as it provides a simple geometric representation of the sine, cosine, and tangent functions. On the unit circle, the angle made with the positive x-axis by a point on the circumference determines the sine and cosine values:

• The x-coordinate of a point on the unit circle is \( \cos(\theta) \).
• The y-coordinate is \( \sin(\theta) \).

Since the circle has a radius of 1, every point on the circle is equidistant from the center, simplifying the representation of angles and their respective trigonometric values. The unit circle also helps determine which quadrants certain trigonometric functions are positive or negative, assisting in solving the initial problem of \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \).

Rationalization

Rationalization is a technique used in algebra to eliminate square roots or complex numbers from the denominators of fractions. It simplifies expressions, making them easier to interpret and further manipulate.
For instance, if you have \( \frac{1}{\sqrt{3}} \), the rationalized form would be \( \frac{\sqrt{3}}{3} \). This is achieved by multiplying both the numerator and the denominator by \( \sqrt{3} \), which removes the square root from the denominator.
This technique is especially useful in trigonometry, as it often pops up when dealing with functions such as cotangent, where preserving exact values is vital for accuracy. Rationalization ensures that numbers and expressions are presented in a standardized and often more digestible form.