Problem 25 - Precalculus: Functions and Graphs

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\) $$\cos \theta=0.8620$$

Step-by-Step Solution

Answer

(a) \(30.76^{\circ}\); (b) \(30^{\circ} 46^{\prime}\).

Step 1: Understand the Problem

We are given the cosine of an angle, \(\cos \theta = 0.8620\), and we need to approximate the acute angle \(\theta\) in two units: to the nearest \(0.01^{\circ}\) and to the nearest \(1^{\prime}\) (minute). The acute angle means \(\theta\) is between \(0^{\circ}\) and \(90^{\circ}\).

Step 2: Find the Angle in Degrees

Using a scientific calculator or trigonometric table, find the value of \(\theta\) such that \(\cos \theta = 0.8620\). The calculator gives \(\theta \approx 30.7623^{\circ}\).

Step 3: Approximate to the Nearest 0.01 Degrees

The angle \(30.7623^{\circ}\) needs to be rounded to the nearest \(0.01^{\circ}\). This results in \(\theta \approx 30.76^{\circ}\).

Step 4: Approximate to the Nearest Minute

First convert \(30.7623^{\circ}\) to minutes. Since 1 degree equals 60 minutes, multiply the decimal part by 60: \(0.7623 \times 60 = 45.738\). This means \(\theta = 30^{\circ} 45.738^{\prime}\). Round \(45.738^{\prime}\) to the nearest minute, which gives \(46^{\prime}\). Thus, \(\theta = 30^{\circ} 46^{\prime}\).

Key Concepts

Acute Angle

An acute angle is any angle that is greater than 0 degrees but less than 90 degrees. These angles are common in trigonometry and are part of most basic geometry problems. Acute angles are easy to remember because they are seen as "sharp" or "pointy" angles. In triangles, an acute angle is usually found in right triangles where one angle is 90 degrees, and the other two angles must then be acute to keep the total at 180 degrees. Believe it or not, all of the angles in an equilateral triangle are also acute, exactly 60 degrees each. Here are some things to remember about acute angles:

• An acute angle is always less than 90 degrees.
• They can be found in various geometric figures, especially triangles.
• For an angle to be an acute angle in solutions, it needs to be within this specified range.

Cosine Function

The cosine function is a fundamental part of trigonometry. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. Denoted as \(\cos(\theta)\), it helps in determining the angle when this ratio is known.

Understanding the cosine function is crucial in calculations involving angular concepts. The value of cosine is maximum when the angle is 0 degrees. As the angle approaches 90 degrees, the cosine value decreases towards zero.

• It is defined for all angles, not just acute ones, but the meaning here focuses on acute angles.
• Cosine values range from -1 to 1 for all angles in its standard form.
• To find the angle from its cosine value, use either a scientific calculator, a cosine table, or inverse cosine functions like \(\cos^{-1}(x)\).

Degree Measurement

Degree measurement is a method of expressing angles. This measurement is one of the most popular because it divides a circle into 360 equal parts. When working with angles, especially in trigonometry, the degree is often preferred due to its simplicity.

Degrees are symbolized by the "°" symbol. Understanding degree measurement helps in seamlessly transitioning from theoretical problems to practical applications, like this exercise where degrees are the required unit of measurement.

• A full circle measures 360 degrees.
• A straight angle measures 180 degrees.
• Angles can also be expressed in smaller units, like minutes (\(1° = 60'\), where the prime symbol represents minutes).

Angle Approximation

Angle approximation is the process of finding a close estimate to the exact angle value. When working with trigonometric calculations, precise measurements are sometimes unnecessary, and approximation provides a practical solution.

With angles, rounding to the nearest desired degree or minute simplifies computations and aligns with typical measurement tools that may not provide decimal precision.

• In trigonometry problems, angles often need to be approximated to the nearest tenth (0.1), hundredth (0.01), or minute (\(1'\)).
• An approximation helps in smoothly converting decimals into easily understandable degrees and minutes.
• This technique is useful not only in academic settings but also in fields like engineering, navigation, and architecture.