When we look at sine, cosine, and tangent in AS-Level Mathematics, it’s cool to see how these math terms can change based on how we look at them.

1. Right-Angled Triangle Definition

In the classic method using right-angled triangles, we define:

• Sine: ( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} )
• Cosine: ( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
• Tangent: ( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} )

For example, if you have a triangle where ( \theta ) is one of the angles, and the side opposite ( \theta ) is 3 units long while the hypotenuse is 5 units long, then:
( \sin(\theta) = \frac{3}{5} ).

2. Unit Circle Definition

In the unit circle, where the radius is 1, the definitions change a bit to use coordinates:

• Sine: This is the y-coordinate of where the angle’s line meets the circle.
• Cosine: This is the x-coordinate.
• Tangent: You can think of it as ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} ).

For instance, for an angle of ( 30^\circ ), the information from the unit circle tells us:
( \sin(30^\circ) = \frac{1}{2} ) and ( \cos(30^\circ) = \frac{\sqrt{3}}{2} ).
So, ( \tan(30^\circ) = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} ).

3. Graphical Interpretation

When we think about graphs, we can see these functions as wave patterns. Sine and cosine waves go up and down between -1 and 1. On the other hand, the tangent wave has gaps, which shows that it sometimes doesn't have a value.

Knowing these definitions in different ways helps you get a better grip on trigonometry and how to use it in various math problems!