Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\csc 145^{\circ} 45^{\prime}$$

Step-by-Step Solution

Answer

The decimal approximation for \( \csc 145^{\circ} 45^{\prime} \) is approximately 1.7391.

Step 1: Convert Degrees and Minutes to Decimal Degrees

To convert the angle from degrees and minutes to decimal degrees, use the formula: \( \text{Degrees} + \frac{\text{Minutes}}{60} = \text{Decimal degrees} \).Here, \( 145^{\circ} \) and \( 45^{\prime} \) becomes:\[ 145 + \frac{45}{60} = 145.75^{\circ} \].

Step 2: Use the Calculator to Find the Sine

On your calculator, input the angle in decimal degrees (145.75) to find \( \sin 145.75^{\circ} \). Typically, this will require setting your calculator to degree mode first. After computation, you get:\( \sin 145.75^{\circ} \approx 0.57499 \).

Step 3: Calculate the Cosecant

The cosecant function is the reciprocal of the sine function. Therefore, to find \( \csc 145.75^{\circ} \), compute:\[ \csc 145.75^{\circ} = \frac{1}{\sin 145.75^{\circ}} \].Use the sine value computed previously:\[ \csc 145.75^{\circ} \approx \frac{1}{0.57499} \approx 1.7391 \].

Key Concepts

Decimal Degrees

Angles in trigonometry are often expressed in degrees and minutes. However, to perform calculations, it's more convenient to convert these angles into decimal degrees. This requires using a simple conversion formula:
\[ \text{Degrees} + \frac{\text{Minutes}}{60} = \text{Decimal degrees} \]
In the problem provided, the angle of \(145^{\circ} 45^{\prime}\) is converted into decimal degrees as follows:
\[ 145 + \frac{45}{60} = 145.75^{\circ} \]
Once converted, the angle is easier to work with in calculators, making computations using trigonometric functions more straightforward.

• Example: \(30^{\circ} 30^{\prime}\) becomes \(30 + \frac{30}{60} = 30.5^{\circ}\).

This approach simplifies handling angles in mathematical computations, especially when a high degree of precision is needed.

Cosecant

The cosecant function, written as \(\csc\), is one of the fundamental trigonometric functions. It is the reciprocal of the sine function. If you understand \(\sin(\theta)\), then
\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]
In the exercise, after converting the angle to decimal degrees, we first find \(\sin 145.75^{\circ} \). That's calculated as approximately \(0.57499\). Using this, the cosecant can be calculated by taking the reciprocal:
\[ \csc 145.75^{\circ} \approx \frac{1}{0.57499} \approx 1.7391 \]
This process highlights how the cosecant function depends directly on the sine. It's practical for solving problems that require the use of the reciprocal of sine, especially in geometry and calculus.

Calculator Usage

Learning how to use a scientific calculator efficiently is vital for solving trigonometric problems. For the provided exercise, ensure that your calculator is set to degree mode since the angle is given in degrees. This step avoids errors and incorrect solutions.
When computing trigonometric functions:

• Input the angle directly if it's already in decimal form, like \(145.75^{\circ}\).
• Use the \(\sin\) button to calculate sine, and if needed, use the reciprocal function or simply divide 1 by the sine result to find cosecant.

Modern calculators will often allow you to perform these operations in succession, displaying results with as many digits of precision as your calculator can handle, which is crucial for homework accuracy. Practicing these steps enhances efficiency and understanding of trigonometric calculations.