Problem 25 - 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. $$\cos 45^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \cos 45^{\circ} \) is \( \frac{\sqrt{2}}{2} \), approximately 0.7071.

Step 1: Recognize the Trigonometric Function

The problem requires us to find the cosine of an angle measured in degrees. Specifically, we need to evaluate \( \cos 45^{\circ} \). This step involves identifying the trigonometric function and angle in the problem.

Step 2: Determine the Exact Value of \( \cos 45^{\circ} \)

The angle 45° is a special angle in trigonometry, for which exact trigonometric values are well known. For \( 45^{\circ} \) (or \( \frac{\pi}{4} \) radians), the exact value of the cosine function is \( \frac{\sqrt{2}}{2} \). This is based on the properties of an isosceles right triangle where the lengths of the legs are equal.

Step 3: Check If the Exact Value is Irrational

The value \( \frac{\sqrt{2}}{2} \) is an irrational number because \( \sqrt{2} \) is irrational. Hence, the exact expression involves an irrational value.

Step 4: Find a Decimal Approximation Using a Calculator

Now, use a calculator to approximate \( \frac{\sqrt{2}}{2} \). When you enter this into a calculator, it yields approximately 0.7071. This step helps verify our understanding and provides a decimal approximation to support the exact value found in step 2.

Key Concepts

Exact Value

In trigonometry, finding the exact value of trigonometric functions for certain angles is an essential skill. For instance, we were tasked with finding the exact value of \( \cos 45^{\circ} \). This particular angle is one of the special angles with well-documented exact values. The exact value of \( \cos 45^{\circ} \) comes from the geometry of a 45°-45°-90° isosceles right triangle. In such a triangle, both legs are of equal length, and this leads to each angle other than the right angle being 45°. The cosine of 45° (or \( \frac{\pi}{4} \) radians) corresponds to the ratio of the length of one of these equal legs to the hypotenuse. By using the Pythagorean theorem, one can derive that this ratio is \( \frac{\sqrt{2}}{2} \). This is considered the exact value because it directly results from trigonometric identities and geometric principles that apply to these special angles. Exact values are useful because they offer precision that a rounded decimal value cannot provide.

Special Angles

Special angles in trigonometry, like 30°, 45°, and 60°, have exact trigonometric values that are commonly used in both academic exercises and practical applications. These angles are heavily emphasized in mathematics due to their frequent occurrence in problems. Recognizing these angles can help simplify calculations significantly.The 45° angle, equivalent to \( \frac{\pi}{4} \) radians, is particularly useful because it forms part of a fundamental set of angles in the study of trigonometric identities. For each of these special angles, exact values for sine, cosine, and tangent can be directly referenced using triangles or the unit circle. Learning these values by heart is advantageous as it allows students to quickly find solutions to problems without needing a calculator. It also lays the groundwork for understanding concepts like angle transformations and trigonometric identities.

Decimal Approximation

While exact values in trigonometry provide theoretical precision, there are instances where using a decimal approximation is practical, particularly when dealing with irrational numbers. In the case of \( \cos 45^{\circ} \), the exact value is \( \frac{\sqrt{2}}{2} \), which is irrational. To support the exact value and make it easier to work with in practical terms, we can find its decimal approximation using a calculator. When computed, \( \frac{\sqrt{2}}{2} \) is approximately 0.7071. This rounded number provides a more digestible form that can be used where approximate numerical values are sufficient.It's important to remember that decimal approximations, while useful, do not provide the exact precision of their original form. However, they are indispensable in fields where precision is secondary to comprehensibility or in situations where digital technology needs to perform computations.