91Ó°ÊÓ

Sine and cosine are fundamental trigonometric functions. They describe the ratios of sides in a right-angled triangle. For an angle \theta in a right triangle:
The sine function is defined as: \ \ \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
The cosine function is defined as: \ \ \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
Understanding these definitions is crucial for solving trigonometric problems.

The tangent function is another key trigonometric function. It relates to both sine and cosine.
It is defined as: \ \ \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
This means that tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle.
An important identity to remember is \(\tan(-\theta) = -\tan(\theta)\).
This property often helps in solving trigonometric expressions.

Simplifying trigonometric expressions often involves rewriting them using basic identities and simplifying step by step.
For example, given the expression \(\frac{1+\tan(-\theta)}{\tan(-\theta)}\), start by using the identity \(\tan(-\theta) = -\tan(\theta)\).
Substitute this into the expression:
\ \ \(\frac{1+(-\tan(\theta))}{-\tan(\theta)} = \frac{1 - \tan(\theta)}{-\tan(\theta)}\)
Next, write \(\tan(\theta)\) in terms of sine and cosine: \ \ \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
Substitute into the expression:
\ \ \(\frac{1 - \frac{\sin(\theta)}{\cos(\theta)}}{-\frac{\sin(\theta)}{\cos(\theta)}} = \frac{\cos(\theta) - \sin(\theta)}{-\sin(\theta)}\)
Finally, simplify to get: \(1 - \cot(\theta)\).
Mastering these steps makes trigonometric simplifications more manageable.