Problem 11 - Algebra and Trigonometry

Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator. $$ \sec 45^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \sec 45^{\circ} \) is \( \sqrt{2} \).

Step 1: Understanding Secant

The secant of an angle \( \theta \) in a right-angled triangle is given by the formula \( \sec(\theta) = \frac{1} {\cos(\theta)} \). In this case, the angle \( \theta \) is 45 degrees. So, to find \( \sec(45) \), you should find \( \cos(45) \) first.

Step 2: Finding the Cosine Value

The cosine of 45 degrees is \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \). These two versions are equivalent, but the second version doesn't have a square root in the denominator because it has been rationalized.

Step 3: Calculating the Secant

Now, substitute the value of \( \cos(45) \) into the formula from Step 1. This will give you \( \sec(45) = \frac{1} {\frac{\sqrt{2}}{2}} = \frac{2} {\sqrt{2}} \).

Step 4: Rationalizing the Denominator

To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} \). So, \( \frac{2}{\sqrt{2}} = \frac{2 \cdot \sqrt{2}}{2} = \sqrt{2} \).

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions. It is denoted as \( \sec \theta \) and is defined as the reciprocal of the cosine function. This means for any angle \( \theta \), the secant is calculated using the formula:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

In a right-angled triangle, the cosine function describes the ratio of the length of the adjacent side to the hypotenuse. Secant, therefore, describes the ratio of the hypotenuse to the adjacent side. This function is particularly useful when solving problems that involve turning cosine relationships into expressions where the hypotenuse is sought after.
For the specific case of 45 degrees, knowing the value of the secant can be very helpful because it belongs to a special triangle, characterized by its symmetrical properties. Understanding secant ensures a broader comprehension of how trigonometric ratios are interrelated.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical process used to eliminate square roots from the denominator of a fraction. This is often a necessary step in math to make expressions simpler and more aesthetically pleasing. When faced with a denominator containing a square root, you multiply both the numerator and denominator by the same square root.
For example, to rationalize \( \frac{2}{\sqrt{2}} \), you multiply both the top and bottom by \( \sqrt{2} \); resulting in

• \( \frac{2 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \= \frac{2 \sqrt{2}}{2} \)

The simplification leads to a result without a square root in the denominator, giving you \( \sqrt{2} \). This step not only simplifies the expression but also makes it consistent with mathematical convention, providing a clearer and more straightforward result.

Cosine of 45 Degrees

The cosine of 45 degrees is a well-known trigonometric value owing to its appearance in the isosceles right triangle. In such a triangle, the two non-hypotenuse sides are of equal length, meaning the angles are all either 45 degrees or 90 degrees.
The value of the cosine for a 45-degree angle is \( \frac{1}{\sqrt{2}} \), but it can also be expressed in a rationalized form as \( \frac{\sqrt{2}}{2} \).

• The conversion between these forms involves multiplying the numerator and the denominator by \( \sqrt{2} \) to remove the square root from the denominator.
• This makes it easier for use in further calculations, especially when finding the secant, which as we've seen involves taking the reciprocal of cosine.

Knowing the cosine of 45 degrees is crucial in trigonometry, as it serves as the basis for understanding more complex trigonometric concepts and calculating functions for similar angles.