Graphs Study Guide

Trigonometric functions are fundamental to understanding periodic phenomena and have wider applications beyond right-angled triangles that you might have encountered before. For angles beyond 90 degrees, we extend the definitions of sine, cosine, and tangent using the unit circle. The unit circle is a circle with a radius of 1 unit, centred at the origin (0,0) of a coordinate plane. An angle $\theta$ is measured counter-clockwise from the positive x-axis. A point P on the unit circle corresponding to angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$. The tangent of $\theta$ is defined as $\frac{\sin\theta}{\cos\theta}$.

This framework allows us to understand the behaviour of these functions for angles between 0 and 360 degrees (or beyond). The graphs of $y = \sin x$, $y = \cos x$, and $y = \tan x$ reveal their periodic nature. The $\sin x$ and $\cos x$ graphs are waves oscillating between -1 and 1, with a period of 360 degrees. The $\tan x$ graph has asymptotes at $90^\[circ]$, $270^\[circ]$ (and so on) and a period of 180 degrees.

Key identities like $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \frac{\sin\theta}{\cos\theta}$ are essential for simplifying expressions and solving trigonometric equations. When solving trigonometric equations, remember that there are often multiple solutions within a given range due to the periodic nature of the functions. Always find the primary solution using your calculator (inverse trig function) and then use the symmetry and periodicity of the graphs or the 'CAST' diagram to find all other solutions within the specified interval, typically $0^\[circ] \leq \theta < 360^\[circ]$. For example, if $\sin\theta = k$, the primary solution is $\theta_1 = \arcsin(k)$. The second solution in the range $0^\[circ] \leq \theta < 360^\[circ]$ for positive $k$ is $\theta_2 = 180^\[circ] - \theta_1$. For negative $k$, the primary solution might be negative, so adjust accordingly by adding 360 degrees. The 'CAST' diagram helps visualise the signs of sine, cosine, and tangent in each quadrant: All (A) positive in Q1, Sine (S) positive in Q2, Tangent (T) positive in Q3, Cosine (C) positive in Q4.

The equation of a circle is fundamental in coordinate geometry, allowing us to describe circular shapes using algebraic expressions. For a circle with its centre at the origin (0,0) and a radius 'r', the equation is simply \(x^2 + y^2 = r^2\). This comes directly from the distance formula (or Pythagoras' theorem), where any point \((x,y)\) on the circle is a distance 'r' from the origin. When the circle's centre is not at the origin, say at \((a,b)\), the equation becomes \((x-a)^2 + (y-b)^2 = r^2\). Here, 'a' represents the x-coordinate of the centre and 'b' represents the y-coordinate of the centre. The terms \((x-a)\) and \((y-b)\) represent the horizontal and vertical distances from the centre to any point on the circle, respectively. Therefore, \((x-a)^2 + (y-b)^2\) is the square of the distance from the centre to that point, which must equal the square of the radius, \(r^2\). Understanding how to convert between the standard form \((x-a)^2 + (y-b)^2 = r^2\) and the expanded general form \(x^2 + y^2 + Dx + Ey + F = 0\) is crucial for higher-level problems. This transformation often involves the technique of 'completing the square' to identify the centre and radius from an expanded equation. Conversely, if you know the centre and radius, you can expand the standard form to get the general form. Problems may also involve finding the equation of a tangent to a circle, which requires understanding the relationship between the radius (perpendicular to the tangent at the point of contact) and the gradient of the tangent, as well as using properties like perpendicular gradients \((m_1 m_2 = -1)\).