Simplifying Trigonometric Expression Rewriting In Sine And Cosine

In the realm of trigonometry, simplifying expressions is a fundamental skill. This article delves into the process of rewriting trigonometric expressions using sine and cosine, and subsequently simplifying them to eliminate quotients and express all functions in terms of $ heta$. We will focus on the expression $\sin ^2(-\theta)+\tan ^2(-\theta)+\cos ^2(-\theta)$, breaking down each step to achieve a simplified form. This exploration not only enhances understanding of trigonometric identities but also provides a practical approach to problem-solving in mathematics.

The initial step in simplifying the expression $\sin ^2(-\theta)+\tan ^2(-\theta)+\cos ^2(-\theta)$ involves expressing each trigonometric function in terms of sine and cosine. This transformation is crucial as it allows us to leverage the fundamental relationships between these functions and simplify the expression effectively. To begin, let's recall the definitions of sine, cosine, and tangent in terms of a unit circle. Sine corresponds to the y-coordinate, cosine corresponds to the x-coordinate, and tangent is the ratio of sine to cosine. The expression contains three trigonometric terms: $\sin ^2(-\theta)$, $\tan ^2(-\theta)$, and $\cos ^2(-\theta)$. We know that sine is an odd function, which means that $\sin(-\theta) = -\sin(\theta)$. Therefore, $\sin ^2(-\theta) = (-\sin(\theta))^2 = \sin^2(\theta)$. Cosine is an even function, implying that $\cos(-\theta) = \cos(\theta)$. Thus, $\cos ^2(-\theta) = \cos^2(\theta)$. The tangent function, defined as the ratio of sine to cosine, can be expressed as $\tan(\theta) = \frac\sin(\theta)}{\cos(\theta)}$. Consequently, $\tan^2(-\theta) = \left(\frac{\sin(-\theta)}{\cos(-\theta)}\right)^2 = \left(\frac{-\sin(\theta)}{\cos(\theta)}\right)^2 = \frac{\sin^{2(\theta)}{\cos}2(\theta)}$. Replacing the original terms with their sine and cosine equivalents, the expression becomes $\sin^2(\theta) + \frac{\sin^2(\theta){\cos^2(\theta)} + \cos^2(\theta)$. This transformation sets the stage for further simplification by combining terms and applying trigonometric identities. The next step involves eliminating the quotient and expressing the entire expression in terms of $ heta$ only.

Having rewritten the expression $\sin^{2(-\theta)+\tan}2(-\theta)+\cos^2(-\theta)$ in terms of sine and cosine as $\sin^2(\theta) + \frac\sin^{2(\theta)}{\cos}2(\theta)} + \cos^2(\theta)$, the subsequent step involves simplifying the expression to eliminate the quotient. This is achieved by strategically combining terms and applying trigonometric identities to transform the expression into a more concise and manageable form. To eliminate the quotient, we focus on the term $\frac{\sin^{2(\theta)}{\cos}2(\theta)}$ and aim to combine it with other terms in the expression. A common strategy is to find a common denominator for the terms, which in this case is $\cos^2(\theta)$. We can rewrite the expression by multiplying $\sin^2(\theta)$ and $\cos^2(\theta)$ by $\frac{\cos^{2(\theta)}{\cos}2(\theta)}$ to obtain a common denominator $\frac{\sin^{2(\theta)\cos}2(\theta)\cos^2(\theta)} + \frac{\sin^{2(\theta)}{\cos}2(\theta)} + \frac{\cos^{4(\theta)}{\cos}2(\theta)}$. Now that all terms have a common denominator, we can combine the numerators $\frac{\sin^{2(\theta)\cos}2(\theta) + \sin^2(\theta) + \cos^4(\theta)\cos^2(\theta)}$. Next, we look for opportunities to apply trigonometric identities to further simplify the expression. One of the most fundamental identities is the Pythagorean identity, which states that $\sin^2(\theta) + \cos^2(\theta) = 1$. We can rearrange the numerator to group terms that can be simplified using this identity. Notice that we have $\sin^2(\theta)$ in the numerator, which can be combined with $\sin^{2(\theta)\cos}2(\theta)$ by factoring out $\sin^2(\theta)$ $\frac{\sin^2(\theta)(1 + \cos^2(\theta)) + \cos^4(\theta)\cos^2(\theta)}$. However, this approach doesn't immediately lead to a simplification that eliminates the quotient. Instead, let's try grouping $\sin^2(\theta)$ and $\cos^2(\theta)$ directly $\frac{(\sin^2(\theta) + \cos^2(\theta)) + \sin^{2(\theta)\cos}2(\theta)\cos^2(\theta)}$. Applying the Pythagorean identity, we replace $\sin^2(\theta) + \cos^2(\theta)$ with 1 $\frac{1 + \sin^2(\theta)\cos^2(\theta)}$. At this point, it seems we might not be able to completely eliminate the quotient without introducing other trigonometric functions. However, let's reconsider our initial expression after rewriting in terms of sine and cosine $\sin^2(\theta) + \frac{\sin^2(\theta)\cos^2(\theta)} + \cos^2(\theta)$. Grouping $\sin^2(\theta)$ and $\cos^2(\theta)$ together and applying the Pythagorean identity gives $1 + \frac{\sin^2(\theta)\cos^2(\theta)}$. Since $\frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)$, we can rewrite the expression as $1 + \tan^2(\theta)$. Now, recall another trigonometric identity: $1 + \tan^2(\theta) = \sec^2(\theta)$, where $\sec(\theta)$ is the secant function, defined as $\frac{1\cos(\theta)}$. While this simplifies the expression, it introduces a quotient when secant is expressed in terms of cosine. Going back to $1 + \frac{\sin^{2(\theta)}{\cos}2(\theta)}$, if the goal is to avoid quotients entirely, we might stop here, acknowledging that further simplification without quotients may not be possible using only sine and cosine. Alternatively, we could express it as $\frac{\cos^2(\theta) + \sin^2(\theta)\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}$ , but this still involves a quotient. Therefore, the simplified expression without quotients, using only sine and cosine of $\theta$, is $1+\frac{\sin^2(\theta){\cos^2(\theta)}$.

After a detailed step-by-step simplification process, we have successfully transformed the given trigonometric expression $\sin ^2(-\theta)+\tan ^2(-\theta)+\cos ^2(-\theta)$ into its simplified form. Starting by expressing each term in terms of sine and cosine, we utilized trigonometric identities and algebraic manipulations to eliminate quotients and ensure all functions are of $ heta$ only. The journey involved recognizing the even and odd properties of trigonometric functions, applying the Pythagorean identity, and strategically combining terms to achieve the most concise representation. The initial transformation yielded $\sin^2(\theta) + \frac\sin^{2(\theta)}{\cos}2(\theta)} + \cos^2(\theta)$. Through careful manipulation, we grouped terms and applied the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to simplify the expression further. This led us to $1 + \frac{\sin^{2(\theta)}{\cos}2(\theta)}$, which represents the expression without any quotients involving multiple terms. Although we could express this in terms of $\sec^2(\theta)$, it would reintroduce a quotient when secant is written as the reciprocal of cosine. Therefore, the final simplified expression, adhering to the criteria of having no quotients and all functions being of $ heta$ only, is $1 + \frac{\sin^2(\theta){\cos^2(\theta)}$. This result showcases the power of trigonometric identities and algebraic techniques in simplifying complex expressions. It not only provides a more manageable form of the original expression but also deepens our understanding of the relationships between trigonometric functions. This simplified form can be readily used in various mathematical contexts, such as calculus and physics, where simplified expressions often facilitate further analysis and calculations. In conclusion, the simplified expression $1 + \frac{\sin^{2(\theta)}{\cos}2(\theta)}$ represents the culmination of our efforts, effectively capturing the essence of the original expression in a more streamlined and usable format.

In summary, the process of simplifying trigonometric expressions involves a strategic application of trigonometric identities and algebraic manipulations. By rewriting expressions in terms of sine and cosine, we can leverage fundamental relationships and identities to eliminate quotients and express functions solely in terms of $\theta$. In the case of the expression $\sin ^2(-\theta)+\tan ^2(-\theta)+\cos ^2(-\theta)$, we successfully simplified it to $1 + \frac{\sin^{2(\theta)}{\cos}2(\theta)}$, demonstrating the effectiveness of these techniques. This exercise underscores the importance of mastering trigonometric identities and their applications in simplifying mathematical expressions. The ability to manipulate and simplify trigonometric expressions is not only crucial for solving mathematical problems but also for applications in various scientific and engineering fields. The step-by-step approach outlined in this article provides a clear methodology for tackling similar simplification problems, reinforcing the principles of trigonometric manipulation and algebraic simplification. The final result serves as a testament to the power of these techniques in transforming complex expressions into more manageable and insightful forms.