Simplifying Trigonometric Expressions A Step-by-Step Guide

In the realm of mathematics, trigonometric expressions often present themselves in complex forms, requiring simplification to reveal their underlying values or relationships. This article delves into the simplification of a specific trigonometric expression, providing a step-by-step guide that not only elucidates the process but also enhances understanding of the fundamental trigonometric identities and principles involved. We will explore how to manipulate angles, apply trigonometric identities, and combine terms to arrive at a simplified form. Whether you are a student grappling with trigonometry problems or a seasoned mathematician seeking a refresher, this guide offers a comprehensive approach to simplifying trigonometric expressions. Understanding these simplifications is crucial for various applications in physics, engineering, and other fields, making this a valuable skill for anyone working with mathematical models. Let's embark on this journey of simplification, unraveling the intricacies of trigonometric expressions and gaining a deeper appreciation for the elegance of mathematical transformations.

Understanding the Initial Expression

The trigonometric expression we aim to simplify is:

$\sin 110^{\circ} \cdot \cos (-40^{\circ}) + \cos 290^{\circ} \cdot \cos 130^{\circ}$

This expression involves trigonometric functions—specifically sine and cosine—of various angles. The angles range from negative values to values greater than 180 degrees, necessitating the use of trigonometric identities to bring them within a more manageable range. Before diving into the simplification process, it's crucial to understand the properties of these functions and how they behave in different quadrants of the unit circle. Sine and cosine are periodic functions, meaning their values repeat after a certain interval. This periodicity, along with the functions' symmetry, allows us to rewrite angles in equivalent forms, which is the cornerstone of simplifying such expressions. For instance, the cosine function is even, meaning $\cos(-x) = \cos(x)$, a property we will use shortly. Additionally, understanding the relationships between angles in different quadrants helps in converting angles to their reference angles, which are acute angles that simplify calculations. Let's break down each term in the expression and strategize how to simplify them using these properties and identities. This initial assessment is vital for a systematic approach to simplification.

Applying Trigonometric Identities

The first step in simplifying the expression involves applying trigonometric identities to handle the angles and functions. We start by addressing the negative angle and angles greater than 180 degrees. Recall that the cosine function is an even function, which means:

$\cos(-x) = \cos(x)$

Applying this to our expression, we can rewrite $\cos(-40^{\circ})$ as $\cos(40^{\circ})$.

Next, we consider $\cos(290^{\circ})$. Since $290^{\circ}$ is greater than $180^{\circ}$, we can find its reference angle by subtracting it from $360^{\circ}$:

$360^{\circ} - 290^{\circ} = 70^{\circ}$

Thus, $\cos(290^{\circ}) = \cos(360^{\circ} - 70^{\circ})$. In the fourth quadrant, cosine is positive, so we can write:

$\cos(290^{\circ}) = \cos(70^{\circ})$

Now, consider $\cos(130^{\circ})$. We can express this angle as:

$130^{\circ} = 180^{\circ} - 50^{\circ}$

Using the identity $\cos(180^{\circ} - x) = -\cos(x)$, we get:

$\cos(130^{\circ}) = -\cos(50^{\circ})$

Substituting these simplifications back into the original expression, we get:

$\sin 110^{\circ} \cdot \cos 40^{\circ} + \cos 70^{\circ} \cdot (-\cos 50^{\circ})$

This transformation has made the expression more manageable by reducing the angles to their reference angles and applying the even/odd properties of trigonometric functions. The next step involves further simplification using other trigonometric identities, focusing on rewriting angles and applying sum-to-product or product-to-sum formulas.

Further Simplification and Angle Transformations

Continuing with the simplification, we observe $\sin(110^{\circ})$. We can rewrite this using the property $\sin(180^{\circ} - x) = \sin(x)$:

$\sin(110^{\circ}) = \sin(180^{\circ} - 70^{\circ}) = \sin(70^{\circ})$

Now our expression looks like this:

$\sin(70^{\circ}) \cdot \cos(40^{\circ}) - \cos(70^{\circ}) \cdot \cos(50^{\circ})$

Notice that $70^{\circ}$ and $40^{\circ}$, as well as $70^{\circ}$ and $50^{\circ}$, might allow us to use trigonometric identities related to the sum or difference of angles. To explore this, let’s rewrite $\cos(50^{\circ})$ using the complementary angle identity:

$\cos(50^{\circ}) = \sin(90^{\circ} - 50^{\circ}) = \sin(40^{\circ})$

Our expression now becomes:

$\sin(70^{\circ}) \cdot \cos(40^{\circ}) - \cos(70^{\circ}) \cdot \sin(40^{\circ})$

This form is highly suggestive of the sine difference identity, which states:

$\sin(A - B) = \sin(A) \cdot \cos(B) - \cos(A) \cdot \sin(B)$

By recognizing this pattern, we can see that our expression perfectly matches the right-hand side of this identity, where $A = 70^{\circ}$ and $B = 40^{\circ}$.

Applying the Sine Difference Identity

Having identified the pattern matching the sine difference identity, we can now apply it to simplify the expression. Recall the identity:

$\sin(A - B) = \sin(A) \cdot \cos(B) - \cos(A) \cdot \sin(B)$

Our expression is:

$\sin(70^{\circ}) \cdot \cos(40^{\circ}) - \cos(70^{\circ}) \cdot \sin(40^{\circ})$

Comparing this with the identity, we can directly substitute $A = 70^{\circ}$ and $B = 40^{\circ}$:

$\sin(70^{\circ} - 40^{\circ}) = \sin(30^{\circ})$

Thus, the expression simplifies to $\sin(30^{\circ})$. Now, we need to evaluate the value of $\sin(30^{\circ})$. This is a standard trigonometric value that is commonly known.

Evaluating the Final Result

The final step in simplifying the expression is to evaluate $\sin(30^{\circ})$. From the unit circle or trigonometric tables, we know that:

$\sin(30^{\circ}) = \frac{1}{2}$

Therefore, the simplified form of the original expression is $\frac{1}{2}$. This result is a concrete numerical value, demonstrating the power of trigonometric identities in reducing complex expressions to their simplest forms. To summarize, we started with a complex expression involving sines and cosines of various angles, applied trigonometric identities to rewrite the angles and functions, recognized the pattern of the sine difference identity, and finally evaluated the resulting trigonometric function to obtain the simplified value. This process highlights the importance of understanding trigonometric identities and their applications in simplifying mathematical expressions. The final answer, $\frac{1}{2}$, is a clear and concise representation of the original expression's value.

Conclusion

In conclusion, we have successfully simplified the trigonometric expression $\sin 110^{\circ} \cdot \cos (-40^{\circ}) + \cos 290^{\circ} \cdot \cos 130^{\circ}$ to its simplest form, which is $\frac{1}{2}$. This simplification process involved several key steps, each relying on fundamental trigonometric identities and principles. We began by applying even-odd properties and reference angles to rewrite the expression in terms of angles within the first and second quadrants. This transformation made the expression more manageable and revealed a pattern that matched the sine difference identity. By recognizing and applying this identity, we were able to reduce the expression to a single trigonometric function, $\sin(30^{\circ})$. Finally, we evaluated this function to obtain the numerical result. This exercise not only demonstrates the utility of trigonometric identities in simplifying complex expressions but also underscores the importance of a systematic approach to problem-solving in mathematics. The ability to manipulate trigonometric expressions is crucial in various fields, including physics, engineering, and computer science. By mastering these techniques, one can tackle more complex problems and gain a deeper appreciation for the elegance and power of mathematics.