Understanding Trigonometric Identities and Graphs

Trigonometric identities, such as the one presented, can sometimes be complex and confusing. In this article, we will clarify the confusion and analyze the given identities and their graphical representations. We will use LATEX to make the expressions more readable and understandable.

Clarifying the Identities

The expression you provided:

dfrac{1 - sin A}{cos A} dfrac{1 - sin A}{cos A} dfrac{sin A}{sin A} dfrac{sin A - sin^2 A}{sin A cos A}

translates to:

(frac{1 - sin A}{cos A} frac{1 - sin A}{cos A} cdot frac{sin A}{sin A} frac{sin A - sin^2 A}{sin A cos A})

This expression simplifies as follows:

First, (frac{sin A}{sin A} 1). Thus, the product (frac{1 - sin A}{cos A} cdot 1) is simply (frac{1 - sin A}{cos A}). Second, the expression simplifies to (frac{sin A - sin^2 A}{sin A cos A}).

However, this expression further simplifies to:

(frac{sin A - sin^2 A}{sin A cos A} frac{sin A(1 - sin A)}{sin A cos A} frac{1 - sin A}{cos A})

It's clear that the initial equality holds true only if the simplification is done correctly.

Graphical Interpretation and Analysis

Let's analyze the given graphs using an angle range of -360° to 360°.

Graph of y1 y2

Period: 360° Magnitude: ∞ and -∞ Representation: Blue line and dots

For the equation y1 y2, we see that there are vertical asymptotes, which means the function can approach ∞ or -∞. This usually indicates a cotangent (or similar) function with vertical asymptotes at regular intervals of 360°.

Graph of y3 y4

Period: 180° Magnitude: 0 at 0°, 0.3247595 at 30°, 0 at 90°, -0.3247595 at 150°, 0 at 180° Representation: Green line and dots

The function y3 y4 appears to be a sine function with a phase shift and period of 180°. The zero crossings and the values at specific angles confirm this. The function has values of 0.3247595 at 30° and -0.3247595 at 150°, which corresponds to the sine of 30° and 150° respectively.

Graph of y5

Period: 180° Magnitude: 0 at 0°, 0.5 at 45°, 0 at 90°, -0.5 at 135°, 0 at 180° Representation: Red line

The function y5 exhibits a period of 180° and specific magnitudes at key angles. This suggests a cosine function with the same period and specific values at 45° and 135°.

Conclusion

To summarize, the trigonometric identities provided were checked for correctness, and their graphical interpretations were analyzed. The given expressions and graphs provided insight into the behavior of trigonometric functions over a range of angles.