Revise Trigonometry for Cambridge 9709 (1.5)

While degrees are convenient for everyday geometry, radians are the standard unit for angles in higher mathematics, especially calculus. A radian is defined based on the properties of a circle itself, making many advanced formulas simpler and more elegant. One full circle rotation is $360°$, which corresponds to an arc length equal to the full circumference, $2\pi r$. This leads to the fundamental conversion factor.

$2\pi \text{ radians} = 360° \quad \implies \quad \pi \text{ radians} = 180°$

Consider a circle with its centre at the origin $(0,0)$ and a radius of 1 unit. Let P be a point that moves along the circumference of this circle. The angle $\theta$ is measured anti-clockwise from the positive x-axis to the line segment OP. The coordinates of the point P, $(x, y)$, are defined as $(\cos \theta, \sin \theta)$. This definition works for any angle $\theta$, as the point P can rotate around the circle multiple times in either direction.

From this, we can also define the tangent function. Since $\tan \theta$ in a right-angled triangle is 'opposite over adjacent', on the unit circle this corresponds to the y-coordinate over the x-coordinate. This gives us a crucial identity.

$\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$

The equation of the unit circle is $x^2 + y^2 = 1$. By substituting $x = \cos \theta$ and $y = \sin \theta$, we derive the most important trigonometric identity, often called the Pythagorean identity.

$\cos^2 \theta + \sin^2 \theta \equiv 1$

The graphs of $y = \sin x$, $y = \cos x$, and $y = \tan x$ are fundamental. The sine and cosine graphs are wave-like curves known as sinusoids. They are periodic, meaning they repeat their pattern at regular intervals. The cosine graph is simply a horizontal translation of the sine graph; specifically, $\cos x = \sin(x + \pi/2)$. The tangent graph is also periodic but looks very different, with vertical asymptotes where the function is undefined (i.e., where $\cos x = 0$).