Problem 43 - Algebra and Trigonometry

Question

Use a calculator to find the value of the trigonometric function to four decimal places. $$ \csc 17^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \csc 17^{\circ} \) to four decimal places is 3.4877.

Step 1: Recognizing the Function

Cosecant function, denoted as \( \csc \), is the reciprocal of the sine function. That means, \( \csc \theta = \frac{1}{\sin \theta} \). Here, you are required to calculate the value of \( \csc \) for 17 degrees.

Step 2: Conversion of Degrees to Radians

The calculators understand angles in terms of radians and not degrees. So you will have to convert the degrees to radians. You do this by using the formula for converting degrees to radians: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Thus, \( 17 \) degrees equals \( 17 \times \frac{\pi}{180} \) radians.

Step 3: Calculating the Required Value

Next, you calculate \( \csc \) of the radians calculated above. Remember, \( \csc \theta = \frac{1}{\sin \theta} \). So now, divide 1 by the sine of the radians.

Step 4: Round to Four Decimal Places

Finally, round the result to four decimal places as asked in the question.

Key Concepts

Cosecant

Understanding the cosecant function is key to solving this problem. Cosecant, denoted as \( \csc \), is a trigonometric function that is the reciprocal of the sine function. To compute the cosecant of an angle \( \theta \), you simply divide 1 by the sine of that angle. Hence, the formula is \( \csc \theta = \frac{1}{\sin \theta} \).
This reciprocal relationship means that whenever you need to find the cosecant of an angle, you're essentially looking for how many times the sine value fits into 1. This concept is especially useful because many calculators and trigonometry problems require knowledge of both sine and its reciprocal, cosecant.

• For acute angles in right triangles, the cosecant provides a useful ratio between the hypotenuse and the opposite side.
• It's one of the six core trigonometric functions, alongside sine, cosine, tangent, secant, and cotangent.

Knowing this helps in understanding step one of the solution.

Degree to Radian Conversion

Calculators often use radians as their standard measure for angles. Therefore, when we are given an angle in degrees, it is essential to convert it into radians. The formula to do this conversion is straightforward: you multiply the degree measure by \( \frac{\pi}{180} \).
For example, converting 17 degrees involves this calculation:
\[ 17 \times \frac{\pi}{180} \]
This transformation aligns the input with how calculators interpret angle measures, allowing them to perform trigonometric calculations accurately.

• Degrees and radians are two units of measuring angles, where \( 180^{\circ} \) is equivalent to \( \pi \) radians.
• Understanding this conversion is crucial for many trigonometric calculations and is often one of the first steps when working with angles in various mathematical problems.

Reciprocal of Sine

When we say that cosecant is the reciprocal of sine, it means that it is the inverse of the sine ratio. To find \( \csc \theta \), you calculate \( 1/\sin \theta \). Essentially, this means turning the sine function on its head.
Let's consider \( \sin \theta = a \), then \( \csc \theta = \frac{1}{a} \).
This conversion plays a significant role in trigonometry, especially in solving problems involving right triangles and periodic functions. Knowing how to find and use the reciprocal form helps in differentiating between situations requiring sine, and those requiring its reciprocal.

• The value of the sine function (and therefore the cosecant) depends on the angle \( \theta \).
• This concept is integral for understanding trigonometric identities and transformations.

Rounding Decimals

Rounding is a fundamental aspect of presenting precise solutions, especially in mathematics where results may have long decimal expansions. When you are asked to round to four decimal places, you look at the fifth decimal place to decide if you round up or stay the same.
For instance, if you have a value like 0.12345, you round to 0.1235 because the fifth decimal is 5 or greater, prompting an increase in the fourth decimal.
This rounding process is vital in ensuring clarity and precision.

• Precision is crucial in fields that demand accuracy, such as engineering and science.
• Proper rounding helps avoid the propagation of errors in subsequent calculations, maintaining consistency across your work.

Applying this rounding correctly is the final step in obtaining the desired four-decimal-place accuracy in trigonometric calculations.