Simplifying Trigonometric Expression With Sine And Cosine

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In the realm of trigonometry, simplifying expressions often involves transforming them into forms that are easier to analyze and manipulate. A fundamental technique in this process is to express all trigonometric functions in terms of sine and cosine. This approach leverages the foundational definitions of these functions and their relationships to other trigonometric functions, such as tangent, cotangent, secant, and cosecant. By converting an expression into its sine and cosine components, we can often utilize trigonometric identities and algebraic simplifications to arrive at a more concise and understandable form. This article delves into the process of expressing a given trigonometric expression in terms of sine and cosine and then simplifying it to eliminate quotients and ensure that all functions are of $ heta$ only. The expression we will focus on is: $\sin ^2(-\theta)-\csc ^2(-\theta)+\cos ^2(-\theta)$

Before we dive into the simplification process, it's crucial to have a firm grasp of the basic trigonometric functions and their definitions. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a right triangle to the ratios of its sides. Sine is defined as the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. Additionally, there are three reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. Understanding these relationships is fundamental to expressing trigonometric functions in terms of sine and cosine. For instance, since csc is the reciprocal of sin, we can express csc(θ) as 1/sin(θ). Similarly, sec(θ) can be expressed as 1/cos(θ), and cot(θ) can be expressed as cos(θ)/sin(θ). These reciprocal relationships are essential tools in our simplification process.

Our initial task is to express the given expression, $\sin ^2(-\theta)-\csc ^2(-\theta)+\cos ^2(-\theta)$, solely in terms of sine and cosine. This involves using the definitions and identities that relate other trigonometric functions to sine and cosine. Let's break down the process step-by-step. First, we need to address the negative angle arguments. We know that sine is an odd function, meaning that sin(-θ) = -sin(θ). Therefore, sin²(-θ) = (-sin(θ))² = sin²(θ). Cosine, on the other hand, is an even function, meaning that cos(-θ) = cos(θ). Thus, cos²(-θ) = cos²(θ). Next, we need to deal with the cosecant term. As mentioned earlier, cosecant is the reciprocal of sine, so csc(θ) = 1/sin(θ). Consequently, csc²(θ) = (1/sin(θ))² = 1/sin²(θ). Applying these transformations to our original expression, we get: sin²(θ) - (1/sin²(θ)) + cos²(θ). Now, our expression is entirely in terms of sine and cosine, which sets the stage for the next phase: simplification.

Now that we have expressed the given expression in terms of sine and cosine, the next step is to simplify it. Our expression currently looks like this: $\sin^2(\theta) - \frac{1}{\sin^2(\theta)} + \cos^2(\theta)$. The goal here is to eliminate the quotient and combine terms to arrive at a more simplified form. A key identity that comes into play here is the Pythagorean identity: sin²(θ) + cos²(θ) = 1. We can rearrange our expression to group the sin²(θ) and cos²(θ) terms together: (sin²(θ) + cos²(θ)) - (1/sin²(θ)). Now, we can directly apply the Pythagorean identity to replace sin²(θ) + cos²(θ) with 1. This simplifies our expression to: 1 - (1/sin²(θ)). To eliminate the quotient, we can find a common denominator and combine the terms. The common denominator in this case is sin²(θ). So, we rewrite 1 as sin²(θ)/sin²(θ), giving us: (sin²(θ)/sin²(θ)) - (1/sin²(θ)). Now, we can combine the numerators over the common denominator: (sin²(θ) - 1) / sin²(θ). This expression is now free of quotients involving multiple terms, but we can simplify it further.

To further simplify the expression (sin²(θ) - 1) / sin²(θ), we can use another form of the Pythagorean identity. We know that sin²(θ) + cos²(θ) = 1. Rearranging this identity, we can express cos²(θ) in terms of sin²(θ) as follows: cos²(θ) = 1 - sin²(θ). Multiplying both sides by -1, we get: -cos²(θ) = sin²(θ) - 1. Now, we can substitute -cos²(θ) for (sin²(θ) - 1) in our expression. This gives us: -cos²(θ) / sin²(θ). This expression is much simpler than our previous one, but we can take it one step further. We know that cotangent (cot) is defined as the ratio of cosine to sine: cot(θ) = cos(θ) / sin(θ). Therefore, cot²(θ) = (cos(θ) / sin(θ))² = cos²(θ) / sin²(θ). So, we can rewrite our expression as: -cot²(θ). This is the final simplified form of our original expression. It is expressed entirely in terms of trigonometric functions of θ, with no quotients and only a single trigonometric function.

Let's recap the entire simplification process step-by-step to ensure clarity.

  1. Original Expression: $\sin ^2(-\theta)-\csc ^2(-\theta)+\cos ^2(-\theta)$
  2. Apply Negative Angle Identities: Recognize that sin(-θ) = -sin(θ) and cos(-θ) = cos(θ), so sin²(-θ) = sin²(θ) and cos²(-θ) = cos²(θ). The expression becomes: $\sin^2(\theta) - \csc^2(\theta) + \cos^2(\theta)$
  3. Express in Terms of Sine and Cosine: Replace csc²(θ) with 1/sin²(θ). The expression becomes: $\sin^2(\theta) - \frac{1}{\sin^2(\theta)} + \cos^2(\theta)$
  4. Apply Pythagorean Identity: Group sin²(θ) and cos²(θ) and use the identity sin²(θ) + cos²(θ) = 1. The expression becomes: $1 - \frac{1}{\sin^2(\theta)}$
  5. Eliminate Quotient: Find a common denominator and combine terms: $\frac{\sin^2(\theta) - 1}{\sin^2(\theta)}$
  6. Apply Pythagorean Identity Again: Use the rearranged Pythagorean identity sin²(θ) - 1 = -cos²(θ). The expression becomes: $\frac{-\cos2(\theta)}{\sin2(\theta)}$
  7. Express in Terms of Cotangent: Recognize that cos²(θ)/sin²(θ) = cot²(θ). The expression simplifies to: $- \cot^2(\theta)$

In conclusion, by systematically applying trigonometric identities and algebraic manipulations, we successfully simplified the given expression $\sin ^2(-\theta)-\csc ^2(-\theta)+\cos ^2(-\theta)$ to $- \cot^2(\theta)$. This process highlights the importance of understanding fundamental trigonometric relationships and identities in simplifying complex expressions. Expressing trigonometric functions in terms of sine and cosine is a powerful technique that allows us to leverage these identities effectively. The final simplified form, -cot²(θ), is a concise representation of the original expression, making it easier to analyze and use in further mathematical operations. Mastering these simplification techniques is crucial for success in trigonometry and related fields of mathematics and physics.