Problem 20 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cos 30^{\circ}$$

Step-by-Step Solution

Answer

(a) Exact: \( \frac{\sqrt{3}}{2} \), (b) Approximation: ≈ 0.866

Step 1: Identify the Important Triangle

The cosine values of standard angles can be derived from a 30°-60°-90° triangle. In such a triangle, the sides are in the ratio 1 : \( \sqrt{3} \) : 2.

Step 2: Determine Cosine for 30 Degrees

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. In a 30°-60°-90° triangle, the adjacent side to 30° is \( \sqrt{3} \), and the hypotenuse is 2. Therefore, \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Step 3: Verify if Exact Value is Irrational

The value \( \sqrt{3} \) is irrational, and thus, \( \frac{\sqrt{3}}{2} \) is also irrational because it involves the irrational number \( \sqrt{3} \).

Step 4: Calculate Decimal Approximation

Using a calculator, calculate \( \frac{\sqrt{3}}{2} \). Enter \( \sqrt{3} \) into the calculator to get approximately 1.732, and then divide by 2. This gives a decimal approximation of about 0.866.

Key Concepts

Cosine

Cosine is one of the fundamental trigonometric functions, typically denoted as \( \cos \). It represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

• For an angle \( \theta \), the cosine function is expressed as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Cosine values are frequently used in various fields like physics, engineering, and even computer graphics for calculating angles and designing circuits.
• Standard angles such as 0°, 30°, 45°, 60°, and 90° have specific cosine values that are used to solve trigonometric problems.

For example, in a 30°-60°-90° triangle, the cosine of 30° is calculated as \( \frac{\sqrt{3}}{2} \), a commonly encountered exact value in trigonometry.The value of \( \cos 30^{\circ} \) being \( \frac{\sqrt{3}}{2} \) reflects the properties of the angles within special right triangles, providing precise results for geometric calculations.

Special Right Triangles

Special right triangles are specific types of triangles that have consistent properties, making calculations with them straightforward and reliable. The 30°-60°-90° triangle is one such special triangle:

• The angles in this triangle are always 30°, 60°, and 90°.
• The side lengths are in the consistent ratio of 1 : \( \sqrt{3} \) : 2, a result derived from the properties of an equilateral triangle.
• Each angle corresponds to predictable trigonometric values, like \( \sin 30^{\circ} = \frac{1}{2} \) and \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \).

Understanding special right triangles is crucial for solving various problems because these triangles establish foundational concepts. They are especially handy when working on geometric designs, architectural blueprints, or calculating forces where right angles are prevalent.Knowing the consistent ratios allows for quick problem-solving and eliminates the need for complicated calculations.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction or ratio of two integers. They have non-repeating, non-terminating decimal expansions. Examples include \( \pi \), \( e \), and \( \sqrt{2} \).

• In mathematics, irrational numbers are important because they represent exact solutions in many contexts where rational numbers fall short.
• For instance, the number \( \sqrt{3} \), which appears frequently in trigonometry, is irrational, meaning it cannot be precisely written as a fraction.
• When combined with rational numbers, such as in the expression \( \frac{\sqrt{3}}{2} \), the result remains irrational.

Though calculating with irrational numbers can pose challenges, modern calculators and computers are equipped to handle them efficiently. In practical scenarios, they are often approximated to a few decimal places, such as 1.732 for \( \sqrt{3} \), for use in computations and measurements.