Definitions of Trigonometric Functions

Trigonometry is the study of the relationships between the side lengths and angles of triangles, derived from the Greek words for 'triangle measure'.

In a right-angled triangle, the three primary functions are defined as ratios of specific sides relative to a reference angle $\theta$.

Sine ($\sin \theta$) is the ratio of the side opposite the angle to the hypotenuse: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

Cosine ($\cos \theta$) is the ratio of the side adjacent to the angle to the hypotenuse: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

Tangent ($\tan \theta$) is the ratio of the opposite side to the adjacent side: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$.

In the unit circle, the angle $\theta$ is always measured anticlockwise starting from the positive x-axis ($0^\circ$).

The x-coordinate of a point on the unit circle represents the Cosine of the angle, while the y-coordinate represents the Sine of the angle.

The Tangent of the angle can be calculated as the ratio of the y-coordinate to the x-coordinate: $\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}$.

This coordinate-based definition allows for the calculation of trigonometric values for obtuse angles ($>90^\circ$), reflex angles ($>180^\circ$), and even negative angles.

Check Calculator Mode: Always verify if the problem requires Degrees or Radians before performing calculations; using the wrong mode is a frequent source of lost marks.

Sanity Checks: Remember that for any real angle $\theta$, the values of $\sin \theta$ and $\cos \theta$ must fall between $-1$ and $1$ inclusive. If your result is outside this range, re-check your steps.

Direction Matters: Ensure you are measuring angles anticlockwise from the positive x-axis; clockwise measurements are considered negative angles.

Inverse Functions: When finding an angle from a ratio, use the inverse functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) and consider if the angle should be acute or obtuse based on the context.

Swapping Coordinates: A common error is associating $x$ with sine and $y$ with cosine; remember that $x$ is the horizontal component (Cosine) and $y$ is the vertical component (Sine).

Tangent at $90^\circ$: Students often forget that $\tan \theta$ is undefined when the x-coordinate is $0$ (at $90^\circ$ and $270^\circ$), as you cannot divide by zero.

Radius Assumption: In the unit circle, the radius is always $1$. If a circle has a different radius $r$, the coordinates become $(r \cos \theta, r \sin \theta)$.