Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate to three decimal places. (a) \(\csc 43^{\circ} 40^{\prime}\) (b) \(\sec 0.26\)

Step-by-Step Solution

Answer

(a) 1.454 (b) 1.035

Step 1: Convert Degrees to Decimal

First, convert the angle in (a) from degrees and minutes to decimal degrees. 1 degree is 60 minutes, so 43 degrees and 40 minutes can be expressed as:\[43 + \frac{40}{60} = 43.6667^{\circ}\]

Step 2: Calculate Cosecant for (a)

To find \(\csc 43.6667^{\circ}\), calculate the sine of the angle first and then take its reciprocal:\[\sin(43.6667^{\circ}) \approx 0.6876\]Now, calculate the cosecant:\[\csc 43.6667^{\circ} = \frac{1}{0.6876} \approx 1.454\]

Step 3: Calculate Secant for (b)

For \(\sec 0.26\), find the cosine of the angle and then compute the reciprocal:\[\cos(0.26) \approx 0.9659\]Then calculate the secant:\[\sec(0.26) = \frac{1}{0.9659} \approx 1.035\]

Key Concepts

Cosecant Calculation

The cosecant is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. If you have a specific angle, say \( \theta \), the cosecant of that angle is given by \( \csc(\theta) = \frac{1}{\sin(\theta)} \). This means to determine the cosecant, you must first calculate the sine of the angle and then take its reciprocal.

For example, to find \( \csc 43.6667^{\circ} \), calculate the sine of \( 43.6667^{\circ} \). Let’s say the sine turns out to be approximately 0.6876. The cosecant would then be:

• Inverse of sine: \( \csc 43.6667^{\circ} = \frac{1}{0.6876} \)
• Approximate value: \( 1.454 \)

Working with trigonometric functions often requires precision, so if the sine value isn't accurately calculated, the cosecant will be off by a bit as well. Remember to keep the calculations accurate to maintain precision.

Secant Calculation

The secant is another vital trigonometric function, defined as the reciprocal of the cosine. For an angle \( \theta \), the secant is expressed as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Like with the cosecant, to find the secant, first compute the cosine of the angle, and then take its reciprocal.

Let's consider \( \sec 0.26 \). First, determine the cosine of 0.26, which is approximately 0.9659. The secant is then calculated as follows:

• Inverse of cosine: \( \sec 0.26 = \frac{1}{0.9659} \)
• Approximate value: \( 1.035 \)

It's important to refer to a calculator or trigonometric tables for precise computations, as even a slight discrepancy in the cosine value may significantly affect the secant result.

Decimal Degrees Conversion

Converting degrees and minutes into decimal degrees is a commonly required calculation in trigonometry. The conversion process simplifies calculations and is necessary for input into many mathematical functions.

To convert from degrees and minutes to decimal degrees, remember that one degree equals 60 minutes. Therefore, if you have an angle given as 43 degrees and 40 minutes, you can convert it as follows:

• Convert minutes to degree part: \( \frac{40}{60} = 0.6667 \)
• Add to the degrees: \( 43 + 0.6667 = 43.6667^{\circ} \)

This simplification allows for seamless calculations in trigonometric functions and is essential for ensuring accuracy when plugging angles into trigonometric calculations. Always double-check conversion steps to avoid errors in subsequent calculations.