Definitions of Trigonometric Functions

In a right-angled triangle, the sides are labeled based on their position relative to the angle $\theta$: the Hypotenuse (longest side), the Opposite (side across from $\theta$), and the Adjacent (side next to $\theta$).

The Sine ratio is defined as the length of the opposite side divided by the hypotenuse: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

The Cosine ratio is defined as the length of the adjacent side divided by the hypotenuse: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

The Tangent ratio is defined as the length of the opposite side divided by the adjacent side: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$.

The Unit Circle is a circle with a radius of $1$ centered at the origin $(0, 0)$ on a Cartesian coordinate plane.

For any angle $\theta$ measured anticlockwise from the positive x-axis, the terminal side intersects the circle at a point $P(x, y)$.

On the unit circle, the x-coordinate of the point represents the cosine of the angle ($x = \cos \theta$) and the y-coordinate represents the sine ($y = \sin \theta$).

This definition allows trigonometric functions to be calculated for angles greater than $90^\circ$ or less than $0^\circ$, which is not possible using simple triangle geometry.

Calculator Mode: Always verify if the problem requires Degrees or Radians; using the wrong mode is one of the most frequent causes of lost marks.

Angle Direction: Remember that positive angles are measured anticlockwise from the positive x-axis, while negative angles are measured clockwise.

Sanity Checks: For the unit circle, sine and cosine values must always fall between $-1$ and $1$. If a calculation results in $\sin \theta = 1.5$, an error has occurred.

Inverse Functions: When using $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$ to find an angle, be aware that calculators only provide the 'principal' value; additional solutions may exist in other quadrants.