Problem 18 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \csc 266^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \csc 266^{\circ} \) is approximately \(-1.0055\).

Step 1: Identify the Cofunction Identities

The trigonometric cofunction identity relevant here is for cosecant and secant: \[ \csc(\theta) = \sec(90^{\circ} - \theta) \] We will use this identity to express \( \csc 266^{\circ} \) in terms of its cofunction.

Step 2: Apply the Cofunction Identity

Substitute \( \theta = 266^{\circ} \) into the cofunction identity: \[ \csc 266^{\circ} = \sec(90^{\circ} - 266^{\circ}) \] Calculate the angle: \[ 90^{\circ} - 266^{\circ} = -176^{\circ} \] Thus, \[ \csc 266^{\circ} = \sec(-176^{\circ}) \]

Step 3: Calculate the Secant of the Angle

Use the identity \( \sec(-\theta) = \sec(\theta) \) since secant is an even function:\[ \sec(-176^{\circ}) = \sec(176^{\circ}) \] Now, use a calculator to find \( \sec(176^{\circ}) \): Convert 176 degrees to radians (since calculators often require radians): \[ 176^{\circ} = 176 \times \frac{\pi}{180} = 3.0718 \] Finally, \( \sec(176^{\circ}) = \frac{1}{\cos(176^{\circ})} \). Calculate using a calculator to four decimal places.

Step 4: Use Calculator to Find Value

Calculate \( \cos(176^{\circ}) \) using a calculator: \[ \cos(176^{\circ}) \approx -0.9945 \] Now, calculate the secant:\[ \sec(176^{\circ}) = \frac{1}{\cos(176^{\circ})} \approx \frac{1}{-0.9945} \approx -1.0055 \]

Key Concepts

Cosecant

Cosecant is one of the six fundamental trigonometric functions, often abbreviated as "csc." It is the reciprocal of the sine function. To put it simply, for any angle \( \theta \), the cosecant is defined as follows:  

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

This means if you know the sine of an angle, you can easily find the cosecant by taking the reciprocal.  
Cosecant is particularly interesting because it's undefined for angles where the sine is zero, as division by zero is undefined in mathematics.  
At angles like \(0^{\circ}, 180^{\circ}\), etc., \(\sin(\theta) = 0\), hence \(\csc(\theta)\) becomes undefined.
It’s always a good practice to ensure the angle you're dealing with is not coinciding with such values when working with cosecant.
Additionally, like other trigonometric functions, the values of cosecant are periodic, repeating every \(360^{\circ}\).
This pattern is essential in solving complex problems that involve multiple cycles of the sine wave.

Secant

Secant is another crucial trigonometric function, noted as "sec." It is the reciprocal of the cosine function, defined for an angle \( \theta \) as:  

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

This means once you have the value of \(\cos\), you simply take its reciprocal to get the \(\sec\).
Similar to the cosecant, the secant function is undefined wherever the cosine of an angle is zero.
Examples of such angles are \(90^{\circ}, 270^{\circ}\), etc., where \(\cos(\theta) = 0\), making \(\sec(\theta)\) undefined.  
The secant function is unique due to its cofunction relationship with the cosecant, which is central to solving trigonometric identities.
The Even-Odd properties of secant remind us that \(\sec(-\theta) = \sec(\theta)\), making it easier to work with negative angles. This is beneficial when converting angles to their standard form or in cofunction identity scenarios.
The secant function, with its periodic nature like all trigonometric functions, repeats every \(360^{\circ}\), a critical point to consider in trigonometric problems.

Trigonometric Functions

Trigonometric functions form the backbone of trigonometry, describing relationships between the angles and sides of triangles. The basic six are: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each function has its own unique definition:

• \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
• \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

These functions extend beyond simple angle-side relationships, playing crucial roles in calculus, polar coordinates, complex numbers, and wave phenomena.
Their periodic nature makes them exceptionally useful in mathematical modeling, particularly in physics and engineering.
Understanding each function's behavior and interrelations, like cofunction identities, can profoundly enhance your problem-solving skills in trigonometry.
By mastering the conversions and identities, such as \(\csc(\theta) = \sec(90^{\circ} - \theta)\), you make significant progress in simplifying complex trigonometric expressions.