Problem 66 - Precalculus

Question

Evaluate the following expressions exactly: $$\csc \left(-\frac{4 \pi}{3}\right)$$

Step-by-Step Solution

Answer

\(\csc\left(-\frac{4\pi}{3}\right) = \frac{2\sqrt{3}}{3}\).

Step 1: Identify the Reference Angle

The given angle is \(-\frac{4 \pi}{3}\). Since it is negative, convert it to a positive angle by adding \(2\pi\) to it. \(-\frac{4\pi}{3} + 2\pi = \frac{2\pi}{3}\). This is the positive coterminal angle.

Step 2: Determine the Function's Quadrant and Sign

The angle \(\frac{2\pi}{3}\) is in the second quadrant. In the second quadrant, the cosecant function (reciprocal of sine) is positive.

Step 3: Calculate the Sine of the Reference Angle

The reference angle for \(\frac{2\pi}{3}\) is \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\). We know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Step 4: Find the Cosecant Value

Since \(\csc(x) = \frac{1}{\sin(x)}\), take the reciprocal of the sine value: \(\csc\left(\frac{2\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\).

Step 5: Simplify the Expression

Rationalize the denominator to simplify: \(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\). Therefore, the expression evaluates to \(\csc\left(-\frac{4\pi}{3}\right) = \frac{2\sqrt{3}}{3}\).

Key Concepts

Understanding Cosecant

The cosecant function, represented as \( \csc(x) \), is one of the main trigonometric functions. It is the reciprocal of the sine function. This means that for any angle \( x \), \( \csc(x) = \frac{1}{\sin(x)} \). Cosecant is particularly useful in situations where it is impractical to divide by zero in sine; however, like other trigonometric functions, it is undefined for angles where sine equals zero.

The graph of the cosecant function reveals vertical asymptotes, which occur at every point where sine is zero. Since these points correspond to angles that are integer multiples of \( \pi \), \( \csc(x) \) is undefined there.

• In degrees, the asypmptotes appear at angles like \( 0°, 90°, 180°,... \)
• In radians errors occur at multiples such as \( 0, \pi, -\pi \) and so on

Understanding these restrictions and the reciprocal nature of cosecant is crucial for solving trigonometric problems.

Grasping Reference Angles

A reference angle is the smallest angle that a given angle makes with the x-axis. It provides a way to determine the trigonometric values of larger angles based on smaller, known angles. When you have an angle expressed in radians, such as \( \frac{2\pi}{3} \), finding its reference involves measuring the shortest distance to the x-axis.

To calculate a reference angle

• Subtract the known angle from \( \pi \) or \( 180° \) if it's in the second quadrant.
• If the angle's in the third quadrant, subtract \( \pi \) from the given angle or \( 180° \).
• In the fourth quadrant, subtract \( 2\pi \) or \( 360° \).

For example, if you find that \( \frac{2\pi}{3} \) is your angle, subtract it from \( \pi \) to determine the reference angle. This results in \( \frac{\pi}{3} \), which helps to easily evaluate trigonometric functions.

The Process of Rationalization

Rationalization is a technique used in algebra to eliminate radicals in the denominator of a fraction. In trigonometry, it's often used to simplify expressions to make them more comprehensible, particularly when dealing with the cosecant function.

When faced with an expression like \( \frac{2}{\sqrt{3}} \), you'll want to get rid of the square root in the denominator. To do this, multiply the numerator and the denominator by the same radical present in the denominator. For instance,

• Multiply both by \( \sqrt{3} \) to give \( \frac{2\sqrt{3}}{3} \).

This process maintains the value of the fraction while producing a more manageable form. Rationalization is crucial in mathematics as it leads to cleaner, more standardized solutions.

Exploring Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, essentially representing the same direction or position despite potentially different magnitudes. You can find coterminal angles by adding or subtracting integer multiples of \( 2\pi \) (or \( 360° \)) from the original angle.

For example:

• An angle of \( 30° \) and \( 390° \) are coterminal, as \( 390° = 30° + 360° \).

This ability to shift angles by entire cycles is particularly useful in trigonometry, as it allows calculations with positive angles even when starting with negatives. In the context of the exercise, changing the original negative angle \( -\frac{4\pi}{3} \) by adding \( 2\pi \) gives \( \frac{2\pi}{3} \). This simplifies the problem, making the evaluation of the trigonometric function more straightforward.