Problem 17 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{2}}{2}\), (b) \(-\sqrt{2}\), (c) \(-1\)

Step 1: Simplify the angle term

The angle given is \(-\frac{\pi}{4}\), which is negative. In trigonometry, the functions have periodic behavior. Since cosine and reciprocal functions are even, we can use this property to simplify the calculation. Recognize that \(\cos(-x) = \cos(x)\) and evaluate at \(\frac{\pi}{4}\).

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

The cosine function \(\cos\left(\frac{\pi}{4}\right)\) corresponds to a 45-degree angle. From trigonometry, we know \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). Thus, \(\cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step 3: Determine \(\csc\left(-\frac{\pi}{4}\right)\)

Since \(\csc(x) = \frac{1}{\sin(x)}\) and \(\sin(-x) = -\sin(x)\), we need to find \(\sin\left(\frac{\pi}{4}\right)\). We know \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). So, \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\). Thus, \(\csc\left(-\frac{\pi}{4}\right) = -\sqrt{2}\).

Step 4: Find \(\cot\left(-\frac{\pi}{4}\right)\)

Recall that \(\cot(x) = \frac{1}{\tan(x)}\). Also, \(\tan(-x) = -\tan(x)\). We need to find \(\tan\left(\frac{\pi}{4}\right)\), which is 1. Therefore, \(\tan\left(-\frac{\pi}{4}\right) = -1\). Hence, \(\cot\left(-\frac{\pi}{4}\right) = -1\).

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions that arises from the study of right-angled triangles. It measures the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Mathematically, it is expressed as \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). The function is periodic with a period of \( 2\pi \), which means that it repeats its values every \( 2\pi \). This property allows us to evaluate the cosine of multiple angles.

• The cosine function is an "even function," meaning \( \cos(-\theta) = \cos(\theta) \). This property is particularly useful for simplifying calculations involving negative angles.
• For \( \theta = \frac{\pi}{4} \), the cosine value is known to be \( \frac{\sqrt{2}}{2} \), which simplifies many trigonometric problems involving 45-degree angles.

When evaluating \( \cos\left(-\frac{\pi}{4}\right) \), we apply the even function property: \( \cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) \). Hence, the cosine value remains \( \frac{\sqrt{2}}{2} \).
Understanding these properties can streamline solving problems involving the cosine function.

Cosecant Function

The cosecant function, abbreviated as \( \csc \), is the reciprocal of the sine function. It is expressed mathematically as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Like sine, cosecant has its own set of properties concerning even and odd functions.

• The sine function is an "odd function," meaning \( \sin(-\theta) = -\sin(\theta) \).
• Accordingly, the cosecant function inherits this property, expressed as \( \csc(-\theta) = -\csc(\theta) \).
• For an angle of \( \frac{\pi}{4} \), \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), leading to \( \csc\left(\frac{\pi}{4}\right) = \sqrt{2} \).

The negative sign in \( \csc\left(-\frac{\pi}{4}\right) \) is critical, as \( \csc(-x) \) takes the negative reciprocal. Therefore, \( \csc\left(-\frac{\pi}{4}\right) = -\sqrt{2} \).
Being comfortable with sine and cosecant relationships helps tackle complex trigonometric calculations.

Cotangent Function

Cotangent, denoted as \( \cot \), is another fundamental trigonometric function that represents the reciprocal of the tangent function. Its mathematical expression is \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

• The tangent function, \( \tan \), being an "odd function," gives us \( \tan(-\theta) = -\tan(\theta) \).
• This means \( \cot(-\theta) = -\cot(\theta) \).
• Knowing that \( \tan\left(\frac{\pi}{4}\right) = 1 \), we have \( \cot\left(\frac{\pi}{4}\right) = 1 \), suggesting \( \tan(x) \) is 1 at 45 degrees.

For \( \cot(-x) \), it's crucial to note that the inverted function also takes the negative sign, making \( \cot\left(-\frac{\pi}{4}\right) = -1 \).
Understanding these interrelations of trigonometric functions simplifies evaluating functions for complex angles.

Angle Reduction

Angle reduction, in trigonometry, involves simplifying angles using the periodic properties of trigonometric functions. This process leverages the symmetric properties of these functions:

• Using properties like evenness and oddness (e.g., \( \cos(-\theta) = \cos(\theta) \) or \( \sin(-\theta) = -\sin(\theta) \)), we can convert angles to simpler forms.
• The periodic nature of functions such as cosine \((2\pi)\) or sine \((2\pi)\), allows simplifying angles beyond the basic \(0\) to \(2\pi\) range.

For the function \( \cos(-\theta) \), we applied this symmetry property, reducing it directly to \( \cos(\theta) \).
This technique provides a powerful tool in transforming and solving complex trigonometric problems, minimizing mistakes and improving clarity in calculations.