Problem 31 - Precalculus: Functions and Graphs

Question

Approximate to four decimal places, when appropriate. (a) cot \((\pi / 13)\) (b) csc 1.32 (c) \(\cos (-8.54)\) (d) \(\tan (3 \pi / 7)\)

Step-by-Step Solution

Answer

(a) 4.7535, (b) 1.4592, (c) -0.8135, (d) -1.3764.

Step 1: Understanding the cotangent function

Cotangent, denoted by \( \cot \), is the reciprocal of the tangent function. For an angle \( \theta \), \( \cot(\theta) = \frac{1}{\tan(\theta)} \). We need to evaluate \( \cot(\frac{\pi}{13}) \).

Step 2: Calculate \( \tan(\frac{\pi}{13}) \)

Calculate \( \tan(\frac{\pi}{13}) \) using a calculator to find the angle in radians. This calculation will give a value that is the ratio of the opposite side to the adjacent side in a right triangle.

Step 3: Find \( \cot(\frac{\pi}{13}) \)

Use the result from the previous step to calculate \( \cot(\frac{\pi}{13}) = \frac{1}{\tan(\frac{\pi}{13})} \). The approximate value is 4.7535.

Step 4: Understanding the cosecant function

Cosecant, denoted by \( \csc \), is the reciprocal of the sine function. For an angle \( \theta \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \). We need to evaluate \( \csc(1.32) \).

Step 5: Calculate \( \sin(1.32) \)

Use a calculator to find \( \sin(1.32) \). This calculation uses the angle as radians.

Step 6: Find \( \csc(1.32) \)

Use the result from the previous step to calculate \( \csc(1.32) = \frac{1}{\sin(1.32)} \). The approximate value is 1.4592.

Step 7: Understanding the cosine function for negative angles

The cosine function is even, which means \( \cos(-\theta) = \cos(\theta) \). We need to evaluate \( \cos(-8.54) \).

Step 8: Evaluate \( \cos(8.54) \)

Use a calculator to find \( \cos(8.54) \). Since cosine is periodic with a period of \( 2\pi \), the angle is converted appropriately.

Step 9: Calculate the cosine value

The cosine value is -0.8135, reflecting the evaluation at the equivalent positive angle.

Step 10: Understanding the tangent function

The tangent function is periodic with a period of \( \pi \). We need to evaluate \( \tan(\frac{3\pi}{7}) \).

Step 11: Calculate \( \tan(\frac{3\pi}{7}) \)

Use a calculator to compute \( \tan(\frac{3\pi}{7}) \) using the angle in radians. The result is the ratio of the opposite side to the adjacent side for the angle in a right triangle.

Step 12: Determine the value of tangent

After evaluating, \( \tan(\frac{3\pi}{7}) \) is approximately -1.3764.

Key Concepts

Cotangent

Cotangent, often written as \( \cot \), is a trigonometric function which serves as the reciprocal of the tangent function. When you encounter \( \cot(\theta) \), it essentially means \( \cot(\theta) = \frac{1}{\tan(\theta)} \). In simpler terms, this means if you know the value of tangent, you can easily find cotangent by taking its reciprocal.

To understand the cotangent function a bit better, imagine a right triangle where \( \theta \) is one of the angles. Tangent of \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side. Thus, cotangent swaps these two dimensions, providing the ratio of the adjacent side to the opposite side.

Let's work through the example provided in the exercise. Here we've got \( \cot(\frac{\pi}{13}) \). Following the method, we first compute \( \tan(\frac{\pi}{13}) \) since cotangent uses tangent. After determining this value, the calculation \( \frac{1}{\tan(\frac{\pi}{13})} \) gives us \( \cot(\frac{\pi}{13}) \), which approximates to 4.7535. This computation is useful in various fields, particularly in engineering and physics for angle determinations.

Cosecant

Cosecant, or \( \csc \), is another trigonometric function that you come across frequently, especially when dealing with complementary functions. It's identified as the reciprocal of the sine function. Simply put, for an angle \( \theta \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

To visualize this with a right triangle, think about the sine of \( \theta \), which is the ratio of the length of the opposite side to the hypotenuse. The cosecant, therefore, "flips" this calculation, making it the hypotenuse to opposite side ratio.

Consider the calculation in the assignment of \( \csc(1.32) \). Using a calculator, we first identify \( \sin(1.32) \). The next step involves evaluating its reciprocal: \( \csc(1.32) = \frac{1}{\sin(1.32)} \). This calculation yields an approximate value of 1.4592. Such manipulations are critical in calculus and geometry where functions must be inverted or manipulated for solutions.

Cosine

Cosine, represented by \( \cos \), is a fundamental trigonometric function that often helps in exploring the relationships within a triangle. It specifically relates to the ratio of the adjacent side to the hypotenuse in a right triangle. Importantly, cosine is known as an even function, which means \( \cos(-\theta) = \cos(\theta) \) for any given angle \( \theta \).

This property of being even is particularly useful when evaluating negative angles. Hence, the exercise required calculating \( \cos(-8.54) \). Thanks to cosine being even, this is as simple as determining \( \cos(8.54) \). After adjusting for the function’s periodic nature (with a period of \( 2\pi \)), a calculator finds the cosine value to be -0.8135.

Understanding cosine's characteristics and functions is helpful not only in geometry but also in diverse fields like physics or engineering, where it naturally appears in wave and oscillation calculations.