Simplifying Trigonometric Expressions A Step By Step Guide

4.1.1 $rac{\cos (360^{\circ}-\theta)(\sin (90^{\circ}+\theta) \sin (-\theta))}{\sin (\theta+180^{\circ})}$

To effectively tackle this trigonometric expression, a foundational understanding of trigonometric identities and properties is indispensable. We will dissect the expression piece by piece, employing trigonometric identities to simplify each component before consolidating them. The primary identities we'll leverage include:

1. Cosine of difference identity: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
2. Sine of sum identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
3. Sine of negative angle: sin(-θ) = -sin(θ)
4. Sine of angle plus 180 degrees: sin(θ + 180°) = -sin(θ)
5. Cofunction identity: sin(90° + θ) = cos(θ)
6. Cosine of co-terminal angles: cos(360° - θ) = cos(θ)

Armed with these identities, we proceed to simplify the expression. The numerator features cos(360° - θ), which simplifies directly to cos(θ) due to the cosine function's periodicity and even symmetry. Similarly, sin(90° + θ) transforms to cos(θ) using the cofunction identity. The term sin(-θ) simplifies to -sin(θ) based on the property of sine being an odd function.

The denominator, sin(θ + 180°), simplifies to -sin(θ) using the sine of angle plus 180 degrees identity. Now, substituting these simplified terms back into the original expression, we get:

$$\frac{\cos (360^{\circ}-\theta)(\sin (90^{\circ}+\theta) \sin (-\theta))}{\sin (\theta+180^{\circ})} = \frac{\cos(\theta) \cdot \cos(\theta) \cdot [-\sin(\theta)]}{-\sin(\theta)}$$

Focusing on simplification, we observe that -sin(θ) appears in both the numerator and the denominator, allowing us to cancel them out, provided that sin(θ) ≠ 0. This leads to:

$$\frac{\cos(\theta) \cdot \cos(\theta) \cdot [-\sin(\theta)]}{-\sin(\theta)} = \cos^2(\theta)$$

Thus, the simplified form of the expression is cos²(θ). This result underscores the importance of recognizing and applying trigonometric identities to streamline complex expressions. By methodically breaking down the original expression and applying appropriate identities, we efficiently arrived at a concise and manageable form. This process not only simplifies the expression but also enhances our comprehension of the relationships between different trigonometric functions and angles.

4.1.2 $\frac{\sin(x-180^{\circ)\cos(x-360}\circ)\tan(-x)}{\sin(-x)\cos(90^\circ + x)}$

To simplify the given trigonometric expression, $\frac{\sin(x-180^{\circ)\cos(x-360}\circ)\tan(-x)}{\sin(-x)\cos(90^\circ + x)}$, we need to utilize various trigonometric identities and properties. This problem showcases the application of several key concepts such as the properties of sine, cosine, and tangent functions in different quadrants, the periodicity of trigonometric functions, and the co-function identities. Each step in the simplification process is crucial to understanding the final result.

Let’s start by simplifying each trigonometric function individually:

1. Simplifying $\sin(x - 180^\circ)$: We can use the identity $\sin(A - B) = \sin A \cos B - \cos A \sin B$. Here, $A = x$ and $B = 180^\circ$. So, $\sin(x - 180^\circ) = \sin x \cos 180^\circ - \cos x \sin 180^\circ$. Since $\cos 180^\circ = -1$ and $\sin 180^\circ = 0$, this simplifies to $\sin(x - 180^\circ) = \sin x (-1) - \cos x (0) = -\sin x$.

2. Simplifying $\cos(x - 360^\circ)$: We know that cosine has a periodicity of $360^\circ$, which means $\cos(x - 360^\circ) = \cos(x)$. This is because subtracting $360^\circ$ does not change the cosine value due to its periodic nature.

3. Simplifying $\tan(-x)$: The tangent function is an odd function, which means $\tan(-x) = -\tan x$.

4. Simplifying $\sin(-x)$: Sine is also an odd function, so $\sin(-x) = -\sin x$.

5. Simplifying $\cos(90^\circ + x)$: Using the co-function identity, $\cos(90^\circ + x) = -\sin x$. This identity is derived from the complementary angle relationships in trigonometry.

Now, substitute these simplified expressions back into the original equation:

$$\frac{-\sin x \cdot \cos x \cdot (-\tan x)}{(-\sin x) \cdot (-\sin x)}$$

Next, we simplify the expression by canceling out terms and rewriting the tangent function. Recall that $\tan x = \frac{\sin x}{\cos x}$. Substituting this in, we get:

$$\frac{-\sin x \cdot \cos x \cdot \left(-\frac{\sin x}{\cos x}\right)}{(-\sin x) \cdot (-\sin x)} = \frac{\sin^2 x \cdot \cos x / \cos x}{\sin^2 x}$$

Notice that $\cos x$ in the numerator cancels out. Thus, we have:

$$\frac{\sin^2 x}{\sin^2 x}$$

Finally, assuming $\sin x \neq 0$, we can cancel $\sin^2 x$ from both the numerator and the denominator, which gives us:

$$1$$

Therefore, the simplified value of the given trigonometric expression is $1$. This comprehensive breakdown demonstrates the step-by-step process of simplifying a complex trigonometric expression using fundamental identities and properties. The ability to manipulate and simplify these expressions is a crucial skill in trigonometry and broader mathematical applications.

In summary, simplifying trigonometric expressions requires a solid grasp of trigonometric identities, properties, and algebraic manipulation. By methodically applying these principles, complex expressions can be transformed into simpler, more manageable forms. The examples discussed above illustrate the power of these techniques in solving trigonometric problems. These skills are not only valuable in academic settings but also in various fields of science, engineering, and computer graphics, where trigonometric functions are frequently used to model and analyze periodic phenomena.