Simplifying Trigonometric Expressions Sin(180°+A) ⋅ Cot(360°-A) ⋅ Cos(2π-A) + Sin²(360°-A)

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Introduction

In this article, we will delve into simplifying a complex trigonometric expression. Trigonometry, a fundamental branch of mathematics, deals with the relationships between angles and sides of triangles. Simplifying trigonometric expressions often involves applying various trigonometric identities and properties. The expression we aim to simplify is: sin(180°+A) ⋅ cot(360°-A) ⋅ cos(2π-A) + sin²(360°-A). This expression combines several trigonometric functions and angle transformations, making it a comprehensive exercise in trigonometric simplification. Understanding the symmetries and periodicities of trigonometric functions is crucial for this task. We will break down each component of the expression, apply appropriate identities, and step-by-step reduce it to its simplest form. The process will not only demonstrate the simplification but also reinforce key concepts in trigonometry.

Understanding Trigonometric Identities

To effectively simplify the given expression, a solid grasp of fundamental trigonometric identities is essential. Trigonometric identities are equations that hold true for all values of the variables involved. These identities enable us to transform and simplify trigonometric expressions. Key identities include the angle sum and difference identities, the cofunction identities, and the Pythagorean identities. For instance, the angle sum and difference identities help us expand expressions like sin(A + B) or cos(A - B). Cofunction identities relate trigonometric functions of complementary angles, such as sin(90° - A) = cos(A). Pythagorean identities, such as sin²(A) + cos²(A) = 1, are crucial for simplifying expressions involving squares of trigonometric functions. Furthermore, understanding the periodicity and quadrant rules of trigonometric functions is vital. Sine, cosine, and tangent functions have specific behaviors in each quadrant of the unit circle. For example, sine is positive in the first and second quadrants, while cosine is positive in the first and fourth quadrants. These rules, combined with the periodic nature of trigonometric functions (e.g., sin(A + 360°) = sin(A)), allow us to simplify expressions involving angles greater than 360° or negative angles. Mastering these identities and rules is the cornerstone of simplifying complex trigonometric expressions.

Step-by-Step Simplification

Let's break down the simplification of the expression sin(180°+A) ⋅ cot(360°-A) ⋅ cos(2π-A) + sin²(360°-A) step by step. This structured approach will make the process clear and easy to follow.

Step 1: Simplify sin(180° + A)

The sine function, sin(180° + A), can be simplified using the sine addition formula. Recall that sin(180° + A) represents the sine of an angle that is 180 degrees plus angle A. From trigonometric identities, we know that: sin(180° + A) = sin(180°)cos(A) + cos(180°)sin(A). Since sin(180°) = 0 and cos(180°) = -1, the expression simplifies to: sin(180° + A) = (0)cos(A) + (-1)sin(A). Therefore, sin(180° + A) = -sin(A). This simplification leverages the properties of sine in different quadrants. When you add 180 degrees to an angle, you essentially reflect the angle across both the x and y axes, which changes the sign of the sine function.

Step 2: Simplify cot(360° - A)

Next, we tackle cot(360° - A). The cotangent function is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). The cotangent function has a period of 180 degrees, meaning cot(360° + x) = cot(x). However, since we have cot(360° - A), we need to consider the effect of the negative angle. We know that cot(-A) = -cot(A). Therefore, cot(360° - A) is equivalent to cot(-A) because adding or subtracting multiples of 360° does not change the cotangent value. Thus, cot(360° - A) = -cot(A). This simplification uses the periodicity and the odd function property of the cotangent function.

Step 3: Simplify cos(2π - A)

The expression cos(2π - A) involves the cosine function. Since 2π radians is equivalent to 360 degrees, we are looking at cos(360° - A). The cosine function is an even function, meaning cos(-A) = cos(A). Therefore, cos(360° - A) = cos(-A) = cos(A). This simplification uses the even function property of the cosine function, which states that the cosine of a negative angle is the same as the cosine of the positive angle. The periodicity of the cosine function also plays a role here, as subtracting 360 degrees from an angle does not change its cosine value.

Step 4: Simplify sin²(360° - A)

Finally, we simplify sin²(360° - A). We know that sin(360° - A) is equivalent to sin(-A) due to the periodicity of the sine function. Since the sine function is an odd function, sin(-A) = -sin(A). Therefore, sin(360° - A) = -sin(A). Now, we need to square this expression: sin²(360° - A) = (-sin(A))². Squaring -sin(A) gives us sin²(A). Thus, sin²(360° - A) = sin²(A). This simplification utilizes both the periodicity and the odd function property of the sine function, as well as the property that squaring a negative value results in a positive value.

Substituting the Simplified Terms

Now that we've simplified each term individually, we can substitute them back into the original expression. The original expression was: sin(180°+A) ⋅ cot(360°-A) ⋅ cos(2π-A) + sin²(360°-A). We simplified each term as follows:

  • sin(180° + A) = -sin(A)
  • cot(360° - A) = -cot(A)
  • cos(2π - A) = cos(A)
  • sin²(360° - A) = sin²(A)

Substituting these simplified terms into the original expression, we get: (-sin(A)) ⋅ (-cot(A)) ⋅ cos(A) + sin²(A). This substitution allows us to work with simpler trigonometric functions and combine them more easily. The next step involves further simplification by using the definitions and identities of trigonometric functions to reduce the expression to its simplest form.

Further Simplification

Let's continue simplifying the expression (-sin(A)) ⋅ (-cot(A)) ⋅ cos(A) + sin²(A). First, we can rewrite the cotangent function in terms of sine and cosine. Recall that cot(A) = cos(A)/sin(A). Substituting this into our expression, we get: (-sin(A)) ⋅ (-cos(A)/sin(A)) ⋅ cos(A) + sin²(A). Now, we can simplify the first term. The -sin(A) and -cos(A)/sin(A) multiply to give cos(A). So, the first term becomes cos(A) ⋅ cos(A), which is cos²(A). Therefore, our expression now looks like: cos²(A) + sin²(A). This simplified expression is a fundamental trigonometric identity. The Pythagorean identity states that cos²(A) + sin²(A) = 1. This identity is a cornerstone of trigonometry and is used extensively in simplifying trigonometric expressions. Therefore, the entire expression simplifies to 1. This final simplification demonstrates the power of using trigonometric identities to reduce complex expressions to their simplest forms.

Final Simplified Form

After the step-by-step simplification process, we've arrived at the final simplified form of the given trigonometric expression. The original expression was: sin(180°+A) ⋅ cot(360°-A) ⋅ cos(2π-A) + sin²(360°-A). Through the application of various trigonometric identities, angle transformations, and algebraic manipulations, we simplified each component of the expression. First, we used the sine addition formula and quadrant rules to simplify sin(180° + A) to -sin(A). Then, we applied the periodicity and odd function property of the cotangent function to simplify cot(360° - A) to -cot(A). For cos(2π - A), we used the even function property of cosine to simplify it to cos(A). Lastly, we simplified sin²(360° - A) to sin²(A) using the periodicity and odd function property of sine. Substituting these simplified terms back into the original expression gave us (-sin(A)) ⋅ (-cot(A)) ⋅ cos(A) + sin²(A). We further simplified this by expressing cot(A) as cos(A)/sin(A), which led to cos²(A) + sin²(A). Finally, applying the Pythagorean identity, cos²(A) + sin²(A) = 1, we arrived at the final simplified form. Therefore, the simplified form of the expression is 1. This result highlights the elegance and interconnectedness of trigonometric identities in simplifying complex expressions.

Conclusion

In conclusion, simplifying the trigonometric expression sin(180°+A) ⋅ cot(360°-A) ⋅ cos(2π-A) + sin²(360°-A) demonstrates the power and utility of trigonometric identities. Throughout this process, we utilized a range of fundamental concepts, including angle sum and difference identities, periodicity of trigonometric functions, even and odd function properties, and the crucial Pythagorean identity. By breaking down the expression into manageable parts and applying the appropriate identities step by step, we successfully reduced it to its simplest form, which is 1. This exercise not only provides a solution to a specific problem but also reinforces the importance of understanding and applying trigonometric identities in simplifying complex expressions. Mastery of these concepts is essential for further studies in mathematics, physics, and engineering, where trigonometric functions frequently appear. The ability to simplify trigonometric expressions efficiently and accurately is a valuable skill for anyone working in these fields. This comprehensive simplification process underscores the beauty and coherence of trigonometric principles.