Graphs Revision Notes

The Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1, centred at the origin $(0,0)$. For any angle $\theta$ (measured anti-clockwise from the positive x-axis), the coordinates of the point $P$ on the unit circle are $(\cos\theta, \sin\theta)$. This definition extends sine and cosine to all angles, including those greater than $90^\[circ]$ or negative. Tangent is defined as $\tan\theta = \frac{\sin\theta}{\cos\theta}$. This means $\tan\theta$ is undefined when $\cos\theta = 0$, i.e., at $90^\[circ]$, $270^\[circ]$, etc. Understanding the unit circle helps visualise the signs of $\sin\theta, \cos\theta, \tan\theta$ in different quadrants and deriving related angle identities.

Trigonometric Graphs and Periodicity

The graphs of $y = \sin x$, $y = \cos x$, and $y = \tan x$ are periodic. $y = \sin x$ and $y = \cos x$ both have a period of $360^\[circ]$ (or $2\pi$ radians) and range from -1 to 1. The $\sin x$ graph starts at $(0,0)$, peaks at $1$ at $90^\[circ]$, crosses the x-axis at $180^\[circ]$, troughs at $-1$ at $270^\[circ]$, and returns to $0$ at $360^\[circ]$. The $\cos x$ graph starts at $(0,1)$, crosses the x-axis at $90^\[circ]$, troughs at $-1$ at $180^\[circ]$, crosses the x-axis at $270^\[circ]$, and returns to $1$ at $360^\[circ]$. The $y = \tan x$ graph has a period of $180^\[circ]$ (or $\pi$ radians) and has vertical asymptotes where $\cos x = 0$, i.e., at $x = 90^\[circ], 270^\[circ], \dots$. Its range is all real numbers.

Key Trigonometric Identities

There are two fundamental trigonometric identities you must know:

1. Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$. This identity comes directly from the unit circle where a right-angled triangle with hypotenuse 1 has adjacent side $\cos\theta$ and opposite side $\sin\theta$. It can be rearranged to $\sin^2\theta = 1 - \cos^2\theta$ or $\cos^2\theta = 1 - \sin^2\theta$.
2. Quotient Identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$. This identity is also derived from the coordinates on the unit circle since $\tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$. These identities are crucial for simplifying trigonometric expressions and solving equations.

Solving Trigonometric Equations

To solve equations like $\sin x = k$, $\cos x = k$, or $\tan x = k$ within a given range (e.g., $0^\[circ] \leq x < 360^\[circ]$), follow these steps:

1. Find the principal value: Use the inverse trigonometric function (e.g., $x = \arcsin(k)$) on your calculator. This usually gives one solution, often in the range $-90^\[circ]$ to $90^\[circ]$ for sine or $0^\[circ]$ to $180^\[circ]$ for cosine.
2. Find other solutions: Use the symmetry of the graphs or the 'CAST' diagram.

• For $\sin x = k$: $x_1 = \arcsin(k)$, $x_2 = 180^\[circ] - x_1$. (General solutions: $x = n360^\[circ] + x_1$ and $x = n360^\[circ] + (180^\[circ] - x_1)$).
• For $\cos x = k$: $x_1 = \arccos(k)$, $x_2 = 360^\[circ] - x_1$. (General solutions: $x = n360^\[circ] \pm x_1$).
• For $\tan x = k$: $x_1 = \arctan(k)$, $x_2 = 180^\[circ] + x_1$. (General solutions: $x = n180^\[circ] + x_1$).

Always ensure your solutions are within the specified range.

Standard Form of a Circle Equation

The standard equation of a circle with centre \((a,b)\) and radius \(r\) is given by \((x-a)^2 + (y-b)^2 = r^2\). This form directly shows the coordinates of the centre, \((a,b)\), and the radius, \(r\). Remember that 'a' is subtracted from x and 'b' is subtracted from y. So, if the equation is \((x+3)^2 + (y-2)^2 = 25\), the centre would be \((-3,2)\) and the radius would be \(\sqrt{25} = 5\).

General Form and Completing the Square

The general expanded form of a circle equation is \(x^2 + y^2 + Dx + Ey + F = 0\). To find the centre and radius from this form, you must use the method of 'completing the square' for both the x-terms and y-terms. For example, to complete the square for \(x^2 + Dx\), you add and subtract \((D/2)^2\) to get \((x + D/2)^2 - (D/2)^2\). Apply this to both variables, then rearrange to match the standard form. This allows you to identify \(a\), \(b\), and \(r^2\), and hence the centre and radius.

Finding the Equation of a Tangent

A tangent to a circle is a line that touches the circle at exactly one point. At the point of contact, the radius is perpendicular to the tangent. This means their gradients are negative reciprocals of each other. To find the equation of a tangent at a point \((x_1, y_1)\) on the circle: 1. Find the gradient of the radius connecting the centre \((a,b)\) to \((x_1, y_1)\). 2. Calculate the gradient of the tangent, which is the negative reciprocal of the radius's gradient. 3. Use the point-slope form of a linear equation, \(y - y_1 = m(x - x_1)\), with the tangent's gradient \(m\) and the point of contact \((x_1, y_1)\) to form the tangent equation.

Intersections of Circles and Lines

To find the points of intersection between a circle and a straight line, substitute the equation of the line into the equation of the circle. This will result in a quadratic equation in one variable (either x or y). Solving this quadratic equation will give you the coordinate(s) of the intersection points. If the quadratic has two distinct real roots, the line intersects the circle at two points. If it has one repeated real root, the line is a tangent. If it has no real roots, the line does not intersect the circle. The discriminant \(b^2 - 4ac\) can be used to determine the number of intersection points.

The Equation of a Straight Line: y = mx + c

The standard form of a straight line equation is \(y = mx + c\). In this equation, 'y' and 'x' are variables representing coordinates on the graph. The 'x' variable is the independent variable, usually plotted on the horizontal axis, and 'y' is the dependent variable, plotted on the vertical axis. Understanding what 'm' and 'c' represent is crucial for interpreting and drawing straight line graphs.

Understanding the Gradient (m)

The gradient, denoted by 'm', measures the steepness and direction of a straight line. It is calculated as the ratio of the vertical change (change in y) to the horizontal change (change in x) between any two points on the line. The formula for the gradient is \(m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\). A positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards. A gradient of 0 indicates a horizontal line, and an undefined gradient indicates a vertical line. Lines with larger absolute gradient values are steeper.

Understanding the y-intercept (c)

The y-intercept, denoted by 'c', is the point where the straight line crosses the y-axis. At this point, the x-coordinate is always 0. Therefore, the coordinates of the y-intercept are (0, c). The value of 'c' tells you where the line starts on the y-axis. If 'c' is positive, the line crosses above the origin; if 'c' is negative, it crosses below; and if 'c' is zero, the line passes through the origin (0,0).

Drawing Straight Line Graphs

To draw a straight line graph from its equation \(y = mx + c\), you need to find at least two points that lie on the line. The easiest way to do this is to choose two different x-values, substitute them into the equation, and calculate their corresponding y-values. For example, choose \(x = 0\) to find the y-intercept (0, c) and then choose another simple value like \(x = 1\) or \(x = 2\) to find a second point. Plot these two points on a coordinate grid and then draw a straight line connecting them, extending it in both directions.

The Equation y = mx + c

The standard form for the equation of a straight line is y = mx + c. In this equation, 'y' and 'x' are variables representing coordinates on the graph. The 'm' represents the gradient of the line, which indicates its steepness and direction. A positive 'm' means the line goes up from left to right, and a negative 'm' means it goes down. The 'c' represents the y-intercept, which is the point where the line crosses the y-axis (when x is 0). For example, in y = 2x + 3, the gradient is 2 and the y-intercept is 3.

Calculating the Gradient (m)

The gradient of a straight line can be calculated using any two points on the line, (x1, y1) and (x2, y2). The formula for the gradient is m = (y2 - y1) / (x2 - x1). This is often remembered as 'change in y over change in x' or 'rise over run'. A horizontal line has a gradient of 0 (e.g., y = 5), and a vertical line has an undefined gradient (e.g., x = 4). Understanding the gradient is key to visualising the slope of the line.

Finding the y-intercept (c)

The y-intercept 'c' is the point where the straight line crosses the y-axis. At this point, the x-coordinate is always 0. If you have the equation y = mx + c, the value of 'c' directly gives you the y-intercept. If you have two points or the gradient and one point, you can substitute the gradient (m) and the coordinates of a point (x, y) into the equation y = mx + c and then solve for 'c'. For example, if y = 3x + c and the line passes through (1, 5), then 5 = 3(1) + c, which means c = 2.

Plotting Straight Line Graphs

To plot a straight line graph from its equation, follow these steps: 1. Choose several x-values (e.g., -2, -1, 0, 1, 2). 2. Substitute each x-value into the equation (y = mx + c) to find the corresponding y-value. This creates a set of coordinate pairs (x, y). 3. Plot these coordinate pairs on a coordinate grid. 4. Draw a straight line connecting all the plotted points, extending it across the grid. Using at least three points helps to ensure accuracy and to catch any calculation errors.