Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate to three decimal places. (a) \(\sec 67^{\circ} 50^{\prime}\) (b) \(\csc 0.32\)

Step-by-Step Solution

Answer

(a) 2.621, (b) 3.178

Step 1: Understanding the problem

To find the approximate values of \( \sec 67^{\circ} 50^{\prime} \) and \( \csc 0.32 \), we first need to convert angles where necessary and use the appropriate trigonometric functions. Remember that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Convert the angle 67°50' to decimal degrees

The given angle is in degrees and minutes. Since 1° equals 60 minutes, we can convert minutes to a fraction of a degree: \( 67^{\circ} 50^{\prime} = 67 + \frac{50}{60} \). Simplifying this, \( 67^{\circ} 50^{\prime} = 67.8333^{\circ} \).

Step 3: Calculate \( \sec 67.8333^{\circ} \)

Use a calculator to find \( \cos 67.8333^{\circ} \), then take the reciprocal to find \( \sec 67.8333^{\circ} \). After calculating, \( \cos 67.8333^{\circ} \approx 0.3816 \), so \( \sec 67.8333^{\circ} = \frac{1}{0.3816} \approx 2.621 \).

Step 4: Calculate \( \csc 0.32 \)

Use a calculator to find \( \sin 0.32 \), which gives \( \sin 0.32 \approx 0.3146 \). Taking the reciprocal, \( \csc 0.32 = \frac{1}{0.3146} \approx 3.178 \).

Step 5: Final approximation

Ensure that both answers are rounded to three decimal places. We conclude that \( \sec 67^{\circ} 50^{\prime} \approx 2.621 \) and \( \csc 0.32 \approx 3.178 \).

Key Concepts

Secant Function

The secant function, noted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). Understanding the secant function is crucial for solving problems involving angles and right triangles.
To calculate the secant of an angle, it is necessary to first find the cosine of that angle. This involves trigonometric table lookups or the use of calculators for precision. Once \( \cos \theta \) is known, simply take the reciprocal to find \( \sec \theta \), which gives how much 'stretched out' the cosine base of an angle is when it extends through a unit circle in trigonometric terms. In real-world scenarios, the secant can be used in optics, architecture, and wherever waveforms and rotations are relevant.

Cosecant Function

The cosecant function, represented as \( \csc \theta \), is similarly defined as the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). It is yet another fundamental aspect of trigonometry used in various applications, such as engineering and physics.
To find the cosecant of an angle, one must first determine the sine of that angle. This can be done using simple trigonometric calculations or with the assistance of calculators. After finding \( \sin \theta \), the cosecant is calculated by taking its reciprocal. The cosecant function is especially significant in the analysis of waves, oscillations, and alternating currents, where it's crucial to understand fluctuating amplitudes.

Angle Conversion

Converting angles from one format to another can often be necessary in trigonometry. Angles can be expressed in various forms such as degrees, radians, or degrees and minutes.
For example, to convert an angle given in degrees and minutes to a decimal degree form, divide the number of minutes by 60 (since one degree equals 60 minutes) and add to the degree value. For example, \( 67^{\circ} 50^{\prime} \) becomes \( 67 + \frac{50}{60} = 67.8333^{\circ} \).
Another common conversion is from degrees to radians. The relationship is given by \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This conversion is especially useful in calculus and higher-level mathematics where calculations often depend on the radian measure due to its natural properties in terms of the unit circle.

Trigonometric Approximation

Trigonometric approximation involves estimating values of trigonometric functions, often to a specified number of decimal places. This is important since many trigonometric functions yield irrational numbers.
To approximate, calculate the trigonometric value using a calculator ensuring the mode is set correctly (degree or radian, depending on the context).

• For example, to find \( \sec 67.8333^{\circ} \), first determine \( \cos 67.8333^{\circ} \) and then compute its reciprocal.
• Similarly, for \( \csc 0.32 \), find \( \sin 0.32 \) first, then take its reciprocal.

These results are then rounded to the required decimal places, such as three decimal places, to ensure precision in applied problems. Such approximations are crucial when dealing with real-world measurements, where absolute values are not practical.