Identifying The Expression With The Least Value A Trigonometric Analysis

In this article, we will delve into the mathematical expressions provided and determine which one has the least value. We will break down each expression step-by-step, utilizing trigonometric identities and simplification techniques to arrive at the final result. This comprehensive analysis will not only reveal the expression with the least value but also enhance your understanding of trigonometric functions and their applications. The expressions we will be analyzing are:

1. ${ \frac{\sqrt{3}(\tan 30^\circ + \sin 60^\circ)}{(\sin^2 30^\circ + \cos^2 30^\circ)(\sec^2 45^\circ - \tan^2 45^\circ)} }$
2. ${ \cos^2 30^\circ + \sec 60^\circ(\sin^4 45^\circ + \tan 45^\circ) }$

Let's embark on this mathematical journey together!

Expression 1: ${ \frac{\sqrt{3}(\tan 30^\circ + \sin 60^\circ)}{(\sin^2 30^\circ + \cos^2 30^\circ)(\sec^2 45^\circ - \tan^2 45^\circ)} }$

To determine the value of the first expression, we need to evaluate the trigonometric functions involved and simplify the expression step by step. This process involves substituting known values and applying trigonometric identities to reduce the expression to its simplest form. Our goal is to break down the complex expression into manageable parts, making it easier to calculate and understand.

First, let's identify the values of the trigonometric functions:

• ${ \tan 30^\circ = \frac{1}{\sqrt{3}} }$
• ${ \sin 60^\circ = \frac{\sqrt{3}}{2} }$
• ${ \sin 30^\circ = \frac{1}{2} }$
• ${ \cos 30^\circ = \frac{\sqrt{3}}{2} }$
• ${ \sec 45^\circ = \sqrt{2} }$
• ${ \tan 45^\circ = 1 }$

Now, substitute these values into the expression:

${\frac{\sqrt{3}(\frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2})}{(\frac{1}{2}^2 + \frac{\sqrt{3}}{2}^2)((\sqrt{2})^2 - 1^2)}}$

Simplify the numerator:

${\sqrt{3}(\frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2}) = 1 + \frac{3}{2} = \frac{5}{2}}$

Simplify the denominator. Recall the Pythagorean identity ${ \sin^2 \theta + \cos^2 \theta = 1 }$, so:

${\sin^2 30^\circ + \cos^2 30^\circ = 1}$

Also,

${\sec^2 45^\circ - \tan^2 45^\circ = (\sqrt{2})^2 - 1^2 = 2 - 1 = 1}$

Thus, the denominator simplifies to:

${(1)(1) = 1}$

Therefore, the first expression simplifies to:

${\frac{\frac{5}{2}}{1} = \frac{5}{2} = 2.5}$

In summary, we started with a complex trigonometric expression and, by substituting known values and applying trigonometric identities, simplified it to a numerical value. This systematic approach is crucial for tackling more complex mathematical problems. The key takeaway here is the importance of knowing trigonometric values and identities, as they are the building blocks for simplifying such expressions. The calculated value for the first expression is 2.5.

Expression 2: ${ \cos^2 30^\circ + \sec 60^\circ(\sin^4 45^\circ + \tan 45^\circ) }$

For the second expression, we follow a similar approach, breaking down the expression into its constituent parts and evaluating each one. Understanding the values of trigonometric functions at specific angles is crucial in this process. We will also utilize algebraic simplification to arrive at the final value. The aim is to meticulously work through the expression, ensuring accuracy at each step.

Let's first recall the values of the trigonometric functions:

• ${ \cos 30^\circ = \frac{\sqrt{3}}{2} }$
• ${ \sec 60^\circ = 2 }$
• ${ \sin 45^\circ = \frac{1}{\sqrt{2}} }$
• ${ \tan 45^\circ = 1 }$

Substitute these values into the expression:

${\cos^2 30^\circ + \sec 60^\circ(\sin^4 45^\circ + \tan 45^\circ) = (\frac{\sqrt{3}}{2})^2 + 2((\frac{1}{\sqrt{2}})^4 + 1)}$

Simplify the terms:

${(\frac{\sqrt{3}}{2})^2 = \frac{3}{4}}$

${(\frac{1}{\sqrt{2}})^4 = (\frac{1}{2})^2 = \frac{1}{4}}$

Substitute these simplified values back into the expression:

${\frac{3}{4} + 2(\frac{1}{4} + 1)}$

Simplify further:

${\frac{3}{4} + 2(\frac{5}{4}) = \frac{3}{4} + \frac{10}{4} = \frac{13}{4}}$

Convert to decimal form:

${\frac{13}{4} = 3.25}$

In summary, the second expression was evaluated by substituting trigonometric values and simplifying. We meticulously worked through each term, ensuring accurate calculations. The result highlights the importance of careful substitution and simplification in mathematical problem-solving. The final value obtained for the second expression is 3.25.

Comparing the Values

Now that we have the values for both expressions, we can compare them to determine which one has the least value. This comparison is a straightforward process, as we simply need to see which numerical result is smaller. Understanding how to compare values is a fundamental skill in mathematics, enabling us to make informed decisions based on quantitative data.

• Expression 1: 2.5
• Expression 2: 3.25

Comparing the two values, we can see that 2.5 is less than 3.25.

Therefore, Expression 1 has the least value.

Conclusion

In this detailed analysis, we evaluated two trigonometric expressions and determined that the first expression, ${ \frac{\sqrt{3}(\tan 30^\circ + \sin 60^\circ)}{(\sin^2 30^\circ + \cos^2 30^\circ)(\sec^2 45^\circ - \tan^2 45^\circ)} }$, has the least value. This involved breaking down each expression, substituting known trigonometric values, and simplifying. The process highlighted the importance of understanding trigonometric identities and applying them systematically to solve complex problems. Mathematical problem-solving is not just about finding the answer, but also about understanding the steps involved and the underlying principles. Through this exercise, we have not only found the solution but also reinforced our understanding of trigonometric functions and simplification techniques.

The key to successfully tackling such problems lies in a methodical approach: first, identify the known values; second, substitute them into the expression; and third, simplify using appropriate identities and algebraic manipulations. This step-by-step method ensures accuracy and helps in understanding the problem better. Remember, practice is crucial in mastering these concepts. The more you practice, the more comfortable you will become with trigonometric functions and their applications.