Problem 19 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\cos \left(-\frac{\pi}{4}\right) \quad\) (b) \(\csc \left(-\frac{\pi}{4}\right) \quad\) (c) cot \(\frac{5 \pi}{3}\)

Step-by-Step Solution

Answer

(a) \( \frac{\sqrt{2}}{2} \); (b) \( -\sqrt{2} \); (c) \( -\frac{1}{\sqrt{3}} \).

Step 1: Understanding Cosine Function

The cosine function is even, which means that \( \cos(-x) = \cos(x) \). Therefore, to find \( \cos \left(-\frac{\pi}{4}\right) \), we can instead find \( \cos \left(\frac{\pi}{4}\right) \).

Step 2: Calculating \( \cos\left(-\frac{\pi}{4}\right) \)

For \( \cos \left( \frac{\pi}{4} \right) \), use the unit circle value: \( \cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \). By the even property of cosine, \( \cos\left( -\frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Step 3: Understanding Cosecant Function

Cosecant is the reciprocal of sine, so \( \csc(x) = \frac{1}{\sin(x)} \). Since sine is an odd function, \( \sin(-x) = -\sin(x) \).

Step 4: Calculating \( \csc\left(-\frac{\pi}{4}\right) \)

First, find \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Thus, \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). The cosecant is \( \csc\left(-\frac{\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \).

Step 5: Understanding Cotangent Function

Cotangent is the reciprocal of tangent, given by \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). To find \( \cot\left(\frac{5\pi}{3}\right) \), determine the cosine and sine at \( \frac{5\pi}{3} \).

Step 6: Calculating \( \cot\left(\frac{5\pi}{3}\right) \)

The angle \( \frac{5\pi}{3} \) is equivalent to \( 2\pi - \frac{\pi}{3} \), lying in the fourth quadrant. Here, \( \cos\left(\frac{5\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2} \). Thus, \( \cot\left(\frac{5\pi}{3}\right) = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} \).

Key Concepts

Even and Odd Functions

In mathematics, functions can often be classified as either even or odd, and these properties help simplify calculations. A function is **even** if substituting

• -x into the function gives the same result as substituting x.
• Mathematically, this is expressed as: \[f(-x) = f(x)\].
• The cosine function is an example of an even function, as seen in the calculated example: \( \cos\left( -\frac{\pi}{4} \right) = \cos\left( \frac{\pi}{4} \right) \).

On the other hand, a function is **odd** if

• - substituting -x into the function returns the negative of the original function value.
• This is expressed as: \[f(-x) = -f(x)\].
• The sine function is a common example of an odd function.
• That's why for \( \sin\left( -\frac{\pi}{4} \right),\) we found \( \sin\left( -\frac{\pi}{4} \right) = -\sin\left( \frac{\pi}{4} \right) \).

Unit Circle

The unit circle is a fundamental concept in trigonometry that's pivotal in understanding how trigonometric functions work. It is a circle with a radius of 1. This circle is centered at the origin of a coordinate plane.

• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

For instance, the point associated with \( \frac{\pi}{4} \) in the unit circle is \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \), giving values for \( \cos\left( \frac{\pi}{4} \right) \) and \( \sin\left( \frac{\pi}{4} \right) \). This makes the unit circle a valuable tool for quickly finding sine and cosine values of angles.
Understanding this circle is crucial,

• especially when working with angles that are multiples of \( \pi \), as those angles often represent key points on the circle.

Reciprocal Functions

Reciprocal trigonometric functions include cosecant, secant, and cotangent, which are reciprocals of sine, cosine, and tangent, respectively.

• **Cosecant (csc):** This is the reciprocal of the sine function, given by \( \csc(x) = \frac{1}{\sin(x)} \).
• **Secant (sec):** The reciprocal of the cosine function, \( \sec(x) = \frac{1}{\cos(x)} \).
• **Cotangent (cot):** This is the reciprocal of the tangent function, \( \cot(x) = \frac{1}{\tan(x)} \), or \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).

Knowing the relationship between these functions is essential because it allows for solving problems involving angles and their trigonometric values more efficiently.
For example, in our original step by step solution, the cosecant at \( -\frac{\pi}{4} \) was calculated as: \( \csc\left( -\frac{\pi}{4} \right) = -\sqrt{2} \), derived by finding the reciprocal of the sine value.

Quadrant Angles

The trigonometric functions have specific signs depending on which quadrant the angle is in. This knowledge is critical for correctly determining the sign of trigonometric values.
The unit circle is divided into four quadrants:

• **First Quadrant:** Sine and cosine are both positive here.
• **Second Quadrant:** Sine is positive, cosine is negative.
• **Third Quadrant:** Sine and cosine are both negative.
• **Fourth Quadrant:** Sine is negative, cosine is positive.

In the original problem, the angle \( \frac{5\pi}{3} \) exists in the fourth quadrant, so we find:

• Cosine remains positive, shown as \( \cos\left( \frac{5\pi}{3} \right) = \frac{1}{2} \).
• Sine is negative, \( \sin\left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2} \).

Having this awareness helps ensure the accuracy of calculated trigonometric function values.