Problem 31 - Algebra and Trigonometry with Analytic Geometry

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) 4.0735, (b) 1.0325, (c) -0.8777, (d) -1.3764

Step 1: Understand the Problem

We need to find the values of specific trigonometric functions: \(\cot(\pi/13)\), \(\csc(1.32)\), \(\cos(-8.54)\), and \(\tan(3\pi/7)\). We shall approximate each result to four decimal places.

Step 2: Calculate \(\cot(\pi/13)\)

Use the identity \(\cot(x) = \frac{1}{\tan(x)}\) to calculate \(\cot(\pi/13)\). First, find \(\tan(\pi/13)\) using a calculator, and then take its reciprocal. \[\tan(\pi/13) \approx 0.2455\]\[\cot(\pi/13) = \frac{1}{0.2455} \approx 4.0735\] (to four decimal places).

Step 3: Calculate \(\csc(1.32)\)

Use the identity \(\csc(x) = \frac{1}{\sin(x)}\) to calculate \(\csc(1.32)\). First, find \(\sin(1.32)\) using a calculator, and then take its reciprocal. \[\sin(1.32) \approx 0.9685\]\[\csc(1.32) = \frac{1}{0.9685} \approx 1.0325\] (to four decimal places).

Step 4: Calculate \(\cos(-8.54)\)

Use the fact that \(\cos(-x) = \cos(x)\) due to cosine's even function property. Calculate \(\cos(8.54)\) using a calculator.\[\cos(-8.54) = \cos(8.54) \approx -0.8777\] (to four decimal places).

Step 5: Calculate \(\tan(3\pi/7)\)

Use a calculator to directly find \(\tan(3\pi/7)\).\[\tan(3\pi/7) \approx -1.3764\] (to four decimal places).

Key Concepts

Understanding Cotangent

The cotangent of an angle is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function. If you know the tangent, you can easily calculate the cotangent.
To get the cotangent, use the formula:

• \( \cot(x) = \frac{1}{\tan(x)} \)

In our exercise, we calculated \( \cot(\pi/13) \). First, compute \( \tan(\pi/13) \) and take its reciprocal to find that \( \cot(\pi/13) \approx 4.0735 \), accurate to four decimal places.
The cotangent is most commonly used in scenarios involving right triangles but is also quite useful in calculus and complex analysis. It's a handy tool when dealing with angles and lengths in various mathematical and real-world problems.

Exploring Cosecant

The cosecant function is another key player in trigonometry, building on the sine function. Like cotangent, cosecant is a reciprocal identity carefully paired with its sine twin.

• The formula is: \( \csc(x) = \frac{1}{\sin(x)} \)

For the exercise, we sought \( \csc(1.32) \). Find \( \sin(1.32) \), then take its reciprocal to obtain \( \csc(1.32) \approx 1.0325 \). Calculating to four decimal places, we've got a precise value.
Understanding cosecant grants deeper insights into wave behaviors and oscillations, especially in physics and engineering contexts. It plays a role in solving problems involving periodic phenomena and signal processing.

Decoding Cosine

The cosine function is well known for its recurring presence with sin in describing oscillatory motion and in transforming coordinate systems. Cosine is an even function—meaning \( \cos(-x) = \cos(x) \), useful for simplifying calculations.
In our exercise, we calculated the value for \( \cos(-8.54) \). Thanks to the even property, \( \cos(8.54) = \cos(-8.54) \approx -0.8777 \), exact to four decimal places.

• Helps determine the adjacent side in triangles.
• Uses many properties in calculus and series expansion.

Cosine forms the backbone of understanding circles, waves, and rotations; key in physics, computer graphics, and simulation problems.

Learning About Tangent

The tangent function is closely related to sine and cosine, defined primarily as the ratio between them. Specifically, tangent measures the slope of an angle—very intuitive for engineers and architects.
You can describe tangent by:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

In the given exercise, \( \tan(3\pi/7) \) was calculated directly to produce \( \approx -1.3764 \), considering four decimal places.
Tangent is indispensable in representing slopes, circular motion, and in trigonometric identities. It's foundational in calculus, where it defines derivatives and integrals of trigonometric functions. This function optimizes solutions across spaces extending from pure mathematics to practical engineering.