Problem 44 - Algebra and Trigonometry

Question

Use a calculator to find the value of the trigonometric function to four decimal places. $$ \sec 55^{\circ} $$

Step-by-Step Solution

Answer

The secant of 55 degrees to four decimal places is 1.7435.

Step 1: Understand the problem

We are required to find the value of the secant of an angle. The secant of an angle is found by taking the reciprocal of the cosine of the angle. In this case, we are looking for the secant of 55 degrees, so we need to find the cosine of 55 degrees and then take the reciprocal of that.

Step 2: Calculate the cosine

First, find the cosine of 55 degrees using a scientific calculator. One thing to be cautious of is to make sure that the calculator is set to degrees, not radians. After setting it to degrees, type '55', then press the 'cos' button. The calculator displays the cosine of 55 degrees.

Step 3: Calculate the secant

Next, find the reciprocal of the cosine of 55 degrees, which is the secant of 55 degrees. The reciprocal of a number is found by dividing 1 by that number. So, divide 1 by the cosine of 55 degrees which will give the secant of 55 degrees.

Step 4: Round to Four Decimal Places

This step asks for the secant of 55 degrees to be presented to four decimal places. This means that when the exact value is calculated to more than four decimal digits, it should be rounded off. If the fifth decimal digit is 5 or above, round up the fourth decimal digit. If it's less than 5, just remove the digits starting from the fifth.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of periodic phenomena, such as waves, and are widely used in various fields like physics, engineering, and even economics.
The primary trigonometric functions include:

• Sine (\( \sin \)): Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (\( \cos \)): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): Represents the ratio of the opposite side to the adjacent side.

In addition to these, there are reciprocal functions like secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)), which extend the functionality and applicability of trigonometric calculations. These functions are pivotal for solving problems that involve angles and periodicity.

Cosine Function

The cosine function is one of the primary trigonometric functions, symbolized as \( \cos \). It is crucial in trigonometry because it provides a relationship between the angle and the lengths of sides in a right triangle. Consider a right triangle, the cosine of an angle \( \theta \) is defined as the ratio of:

• Adjacent Side to
• Hypotenuse

Mathematically, this can be expressed as:\[ \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \]When dealing with angles, it's crucial to ensure calculations are in the correct unit (degrees or radians). For example, when calculating \( \cos 55^{\circ} \), ensure your calculator is set to degrees. Using the cosine function correctly allows us to explore further relationships and properties within triangles and helps solve real-world problems related to wave mechanics and oscillations.

Reciprocal Function

A reciprocal function is a very useful concept in trigonometry, specifically when dealing with the secant function. The reciprocal of a number is simply 1 divided by that number. In trigonometric terms, the secant function, denoted by \( \sec \), is a reciprocal function.
For an angle \( \theta \), it's defined as:\[ \sec \theta = \frac{1}{\cos \theta} \]This means to find the secant of an angle, we first determine its cosine and then take the reciprocal. For example, to find \( \sec 55^{\circ} \), we calculate \( \cos 55^{\circ} \) and then take \( \frac{1}{\cos 55^{\circ}} \). Reciprocal functions help simplify and solve many trigonometric problems and can be vital in solving equations involving angles and wave functions.

Angle Measurement

In mathematics, angles can be measured in different units, the most common being degrees and radians. Understanding these measurements is vital for anyone studying trigonometry, as many calculations depend on the proper configuration of these units.

• Degrees: A complete circle is 360 degrees. Most introductory mathematics and geometry problems are expressed in degrees.
• Radians: A complete circle is \( 2\pi \) radians. Radians are often used in higher mathematics, physics, and engineering contexts.

To switch between these units, conversions are often necessary. For instance, to convert degrees to radians, multiply the degree value by \( \frac{\pi}{180} \). Conversely, for radians to degrees, multiply by \( \frac{180}{\pi} \). Understanding and correctly setting your calculator to the right angle measurement can prevent errors in computations and improve accuracy in solving trigonometric problems.