Understanding Trigonometric Identities: Simplifying sin 90° - A and cos 180° - A

When working with trigonometric functions, it's important to understand how to simplify and manipulate expressions involving angles. This article will focus on the expression (frac{sin90^circ - Acdotcos180^circ - A}{sin180^circ Acdotcos90^circ A}). We will break down the expression, apply relevant trigonometric identities, and simplify it step by step. Let's dive into the details.

Breaking Down the Expression

First, let's break down the expression into smaller parts and apply the relevant trigonometric identities. We will use the following identities:

(sin90^circ - A cos A) (cos180^circ - A -cos A) (sin180^circ A -sin A) (cos90^circ A -sin A)

Simplifying the Expression

Now, let's reassemble these pieces and simplify the expression:

Calculate the numerator:[sin90^circ - A cos A] Calculate the first term in the denominator:[cos180^circ - A -cos A] Calculate the second term in the denominator:[sin180^circ A -sin A] Calculate the denominator: (-cos A cdot -sin A sin Acos A) Combine the numerator and denominator: [frac{cos A - (-cos A)}{sin Acos A} frac{cos A cos A}{sin Acos A} frac{2cos A}{sin Acos A} frac{2}{sin A}]

However, a more straightforward approach is to break it down as follows:

Numerator (sin90^circ - Acdotcos180^circ - A cos Acdot -cos A -cos^2A)

Denominator (sin180^circ Acdotcos90^circ A -sin Acdot -sin A sin^2A)

Now, dividing the numerator by the denominator:

[ frac{-cos^2A}{sin^2A} -cot^2A]

Final Answer

The simplified form of the expression (frac{sin90^circ - Acdotcos180^circ - A}{sin180^circ Acdotcos90^circ A}) is:

[-cot^2A]

Conclusion

Understanding and simplifying trigonometric expressions is crucial in various mathematical and engineering applications. By breaking down the expression and applying trigonometric identities, we can arrive at a simpler form that can be more easily understood and used.

Resources for Further Learning

If you would like to delve deeper into trigonometry and algebraic manipulations, consider exploring these resources:

Khan Academy: Trigonometry Math is Fun: Trigonometry Mathcentre: Trigonometric Functions