Basic Essential Mathematics — JEE Advanced Physics

Introduction to Mathematical Foundations

Why Mathematics in Physics?

Physics describes nature in the language of mathematics. The motion of a planet, the fall of a stone, the current in a wire — each is written as an equation. To read physics, you must first read this language.

This chapter builds that foundation. With it you will be able to:

State physical quantities precisely
Solve physics problems step by step
Predict the outcome of a physical process before it happens

Numbers and the Number Line

Every real number has a place on the number line. In physics the number line is a one-dimensional coordinate system: the position of a car on a straight road is a single number.

A number line used as a one-dimensional coordinate system, with the origin marked and the two directions labelled

Negative numbers and absolute value. In physics a negative sign carries direction. A position of $-15 \text{ m}$ lies on the opposite side of the origin from $+15 \text{ m}$ . The absolute value drops the sign and keeps the size. It answers how much, not which way.

Example. A car moves from position $x_A = +10 \text{ m}$ to position $x_B = -15 \text{ m}$ . Its displacement is $\Delta x = x_B - x_A = -15 - 10 = -25 \text{ m}$ . The minus sign gives the direction of the change. The distance travelled is the magnitude: $|\Delta x| = |-25| = 25 \text{ m}$ .

Order of operations (BODMAS/PEMDAS). Every calculation follows one fixed order:

Brackets / Parentheses
Orders / Exponents (powers and roots)
Division and Multiplication (left to right)
Addition and Subtraction (left to right)

Example. Take the kinematic formula $s = u t + \frac{1}{2} a t^2$ with initial velocity $u = 5 \text{ m/s}$ , time $t = 4 \text{ s}$ , and acceleration $a = 10 \text{ m/s}^2$ :

$s = 5 \times 4 + \frac{1}{2} \times 10 \times 4^2$

Work through it in order:

Exponents: $4^2 = 16$ . The expression becomes $5 \times 4 + \frac{1}{2} \times 10 \times 16$ .
Multiplication, left to right: $5 \times 4 = 20$ , and $\frac{1}{2} \times 10 \times 16 = 80$ . The expression becomes $20 + 80$ .
Addition: $20 + 80 = 100 \text{ m}$ .

If a particle is at position $x_1 = -5 \text{ m}$ and another is at $x_2 = 3 \text{ m}$ , the distance between them is the absolute value of their difference. What is the distance in meters?

Evaluate: $3 + 4 \times 2$

Algebraic Operations and Equations

Variables, Constants, and Expressions

Algebra states a physical rule once, for every case. Instead of saying that a car moving at $10 \text{ m/s}$ for $5 \text{ s}$ travels $50 \text{ m}$ , we write $d = vt$ . The letters stand for the quantities.

Constants and variables. Take Newton's second law, $F = ma$ . For one object of fixed mass, $m$ is a constant, while the applied force $F$ and the resulting acceleration $a$ are variables.

Examples of expressions:

$3x + 5$ (linear expression)
$ut + \frac{1}{2}at^2$ (quadratic expression; the displacement formula)

Simplifying Expressions

Like terms carry the same variable to the same power, and only like terms can be combined.

Example: $3x + 5x - 2x = 6x$

Equations and Isolating Variables

In physics you often know the formula and need one particular variable out of it. Rearranging a formula is therefore a basic skill.

The golden rule. An equation is a balance. Whatever is done to one side — adding, subtracting, multiplying, dividing, squaring, taking a root — must be done to the other. The equality then survives every step.

To take the acceleration out of Newton's second law, $F = ma$ , undo the multiplication by dividing both sides by $m$ :

$\frac{F}{m} = \frac{ma}{m} \implies a = \frac{F}{m}$

Linear equations. A linear equation has highest power 1. Its general form is $ax + b = c$ .

Example. Solve $5x + 3 = 18$ .

Subtract 3 from both sides: $5x = 18 - 3 = 15$
Divide both sides by 5: $x = \frac{15}{5} = 3$

Rearranging a physics formula. From the kinematics equation $v = u + at$ , solve for the acceleration $a$ :

Subtract $u$ from both sides: $v - u = at$
Divide both sides by $t$ : $a = \frac{v - u}{t}$

Solve for $x$ : $2x + 7 = 15$

Ratios, Proportions, and Variations

Physical quantities are often tied to one another in fixed proportion. Recognising the form of the tie — direct, inverse, or inverse square — is half the work of a physics problem.

Ratios. A ratio compares two numbers by division. The diameter of a circle is always twice its radius: $\frac{D}{R} = 2$ . Many physical quantities are themselves defined as ratios:

Resistance: the ratio of voltage to current, $R = \frac{V}{I}$ (Ohm's law).
Density: the ratio of mass to volume, $\rho = \frac{m}{V}$ .

Observe the green curve representing $y = \frac{1}{x}$ in the graph above. As the variable $x$ increases from $1.0$ to $2.0$ , what happens to the value of $y$ ?

Direct Variation ( $y \propto x$ ): As one increases, the other increases in the same proportion. Equation: $y = kx$ .
Inverse Variation ( $y \propto \frac{1}{x}$ ): As one increases, the other decreases. Equation: $y = \frac{k}{x}$ .
Inverse Square Variation ( $y \propto \frac{1}{x^2}$ ): Common throughout physics — gravity and the electric force both follow it.

The gravitational force is inversely proportional to the square of the distance ( $F \propto \frac{1}{r^2}$ ). If you double the distance, what happens to the force?

Boyle's Law states that for a fixed amount of gas at a constant temperature, pressure $P$ is inversely proportional to volume $V$ ( $P \propto \frac{1}{V}$ ). If a gas initially at pressure $P_1$ occupies a cylinder, and you compress the gas so that the volume is reduced to one-third of its original size, what happens to the pressure?

Important Algebraic Identities

These identities shorten many derivations in physics. Open each list below:

Square Identities

Square of a Sum: $(a + b)^2 = a^2 + 2ab + b^2$
Square of a Difference: $(a - b)^2 = a^2 - 2ab + b^2$
Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$
Square of a Trinomial: $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

Cube Identities

Cube of a Sum: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b)$
Cube of a Difference: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a - b)$
Sum of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Product & Advanced Identities

Product of Two Binomials: $(x + a)(x + b) = x^2 + (a + b)x + ab$
Product of Three Binomials: $(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$
Sum of Three Cubes Identity: $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

Compute $103 \times 97$ without long multiplication, using an identity.

If $a + b = 7$ and $ab = 12$ , what is $a^2 + b^2$ ?

Various Functions and Their Graphs

Modeling Relationships

A function is a rule that turns an input into exactly one output. Give the position function a time, and it returns the position at that time. A graph shows the whole rule at once — the entire history of a motion in one picture. The graph below is a first example: the position $x = 2t$ of a walker, read off for any time at a glance.

Graphical Analysis in Physics

A graph carries a full table of data in a single figure: the score of a cricket match over by over, the temperature of the Earth year by year, the inflation rate month by month. One glance gives the trend.

Physics uses graphs the same way. The kinetic energy of a car, $K = \frac{1}{2}mv^2$ , plotted against its speed $v$ , rises as a steepening curve: doubling the speed quadruples the destructive energy in a crash.

The slope and the intercepts of such graphs carry physical meaning — an initial velocity, an acceleration.

Understanding the Slope

The slope of a graph is the rate of change of the vertical quantity with respect to the horizontal one. It tells you how fast one quantity changes as the other advances:

Constant Slope (Straight Lines): A straight line has one slope along its whole length — a uniform rate of change. In a displacement-time graph, a straight diagonal line means motion at constant velocity.
Changing Slope (Curved Lines): On a curve the slope changes from point to point. The slope at any point is the slope of the tangent drawn to the curve there:
- Increasing Slope: The curve steepens from left to right. In a displacement-time graph this means the velocity is growing — acceleration.
- Decreasing Slope: The curve flattens. In a displacement-time graph this means the velocity is falling — deceleration.

The three graphs below show the three cases side by side. In the first, equal times bring equal gains in position. In the second, each second brings a larger gain than the one before. In the third, each second brings a smaller gain, and the curve levels off.

1. Linear Functions

A linear function has a constant rate of change. Its graph is a straight line. The general equation is $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.

The Identity Function: $y = x$

With slope $m = 1$ and intercept $c = 0$ , the function reduces to $y = x$ , the identity function. Every output equals its input. It describes a one-to-one relationship, such as motion at exactly $1\text{ m/s}$ : the displacement in meters equals the time elapsed in seconds.

Changing the Slope: $y = 2x$

Raise the slope to $m = 2$ , keeping $c = 0$ , and the line becomes twice as steep. For every unit change in $x$ , the output $y$ changes by 2 units. In physics this is motion at a higher constant velocity: at $2\text{ m/s}$ , the displacement grows twice as fast as time.

Adding an Offset (y-intercept): $y = x + 5$

Keep the slope $m = 1$ and set the intercept $c = 5$ . The line has the same steepness — it is parallel to $y = x$ — but sits 5 units higher. The intercept is the value of $y$ at $x = 0$ . In physics it is the initial value: motion that starts from $5\text{ m}$ instead of the origin.

Exercise: Match the Line

Adjust the sliders for slope $m$ and intercept $c$ until your blue line lies on the gold target line.

The temperature at 8:00 am is found to be $36^\circ\text{C}$ . It is increasing linearly by $0.5^\circ\text{C}$ every hour. At what time will it be $38^\circ\text{C}$ ?

2. Quadratic Functions

A quadratic function contains the square of the variable and nothing higher. Its graph is a parabola. The general form is $y = ax^2 + bx + c$ . A body in free fall under gravity follows a parabolic trajectory, so parabolas appear throughout mechanics.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of $x$ where the parabola crosses the x-axis (where $y = 0$ ). Depending on the discriminant ( $b^2 - 4ac$ ), a parabola can have different types of roots:

Two distinct real roots: The parabola crosses the x-axis at two different points. For example, $y = x^2 - 4$ crosses at $x = 2$ and $x = -2$ .
One repeated real root: The parabola just touches the x-axis at a single point (its vertex). For example, $y = (x - 2)^2$ touches the x-axis exactly at $x = 2$ .
No real roots: The parabola never crosses the x-axis; it lies entirely above or below it. For example, $y = x^2 + 4$ never reaches $y = 0$ .

Transformations of Parabolas

The position of a parabola shifts when its equation is modified:

Horizontal Shifts: Replacing $x$ with $(x + \Delta x)$ shifts the whole graph horizontally. Counter-intuitively, $y = (x + 2)^2$ shifts the graph to the left by 2 units, while $y = (x - 2)^2$ shifts it to the right by 2 units.
Vertical Shifts: Adding a constant to the whole function shifts it along the y-axis. $y = x^2 + 3$ moves the standard parabola up by 3 units; $y = x^2 - 3$ moves it down by 3 units.

Exercise: Match the Parabola

Adjust the sliders until your blue curve lies on the gold target parabola.

A ball is thrown upwards. Its height $h$ in meters after $t$ seconds is given by the quadratic function $h(t) = -5t^2 + 20t + 2$ . At what time(s) does the ball reach a height of $17 \text{ meters}$ ?

The equation of motion of a particle is given by $y = 3x^2 + 2x + 1$ . Which type of curve does this represent?

3. Exponential Functions

In an exponential function, the independent variable sits in the exponent. The general mathematical form is:

$y = a \cdot r^x + c$

where each parameter controls one feature of the curve:

$a$ (Scale Factor): Controls the vertical stretch, compression, or inversion. If $a > 0$ , the curve stays above the asymptote; if $a < 0$ , the curve is flipped upside down.
$r$ (Base / Growth Rate): Must be a positive number ( $r > 0$ ). It decides whether the curve grows or decays.
$c$ (Vertical Shift / Asymptote): Moves the whole graph up or down. The horizontal line $y = c$ is the horizontal asymptote.

Exponential Growth vs. Decay

The behaviour of the function follows from the base $r$ :

Growth ( $r > 1$ ): The value climbs faster and faster as $x$ increases — the growth of a population, for example.
Decay ( $0 < r < 1$ ): The value drops rapidly and levels off towards the asymptote — the cooling of hot tea, or radioactive half-life.

Exercise: Match the Exponential Curve

Adjust the sliders for scale $a$ , base $r$ , and shift $c$ until your blue curve lies on the gold target curve.

A bacterial colony doubles its population every hour. If the initial population at $t = 0$ is $100$ , which exponential function correctly models the population $P(t)$ after $t$ hours?

Basic Geometry and Graphs

Mapping Space with Geometry

Physics happens in space, and geometry is the tool that maps it. We use it to trace light rays, calculate volumes, and resolve vectors.

Triangles. The Pythagorean theorem and similar triangles do most of the geometric work in this course. Consider a person standing near a street lamp. The lamp, the person, and the light rays form two nested right triangles with the same angles. Comparing the ratios of their sides gives the length of the shadow, or the height of the lamp.

A person standing near a street lamp, showing how the similar triangles are formed between the person, their shadow, and the lamp height.

Coordinate System and Distance

A coordinate system fixes the position of a point in space by numbers.

The straight-line distance d between P1 and P2, built from a right triangle in the Cartesian plane. — The straight-line distance d between $P_1$ and $P_2$ , built from a right triangle in the Cartesian plane.

In a two-dimensional Cartesian plane, the straight-line distance $d$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ follows directly from the Pythagorean theorem, as shown above:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Three dimensions. Real motion happens in three dimensions. A point then needs three coordinates $(x, y, z)$ , with the $z$ -axis perpendicular to the $xy$ -plane. The distance formula gains one term:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Slopes and Line Equations

The slope $m$ of a line through points $P_1$ and $P_2$ gives the rate of change. On the page, it is the steepness of the line.

The slope of a line as rise over run, read off a right triangle between two points.

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

Equations of Lines:

Slope-Intercept Form: $y = mx + c$ , where $m$ is the slope and $c$ is the $y$ -intercept.
Point-Slope Form: $y - y_1 = m(x - x_1)$ , the line through the point $(x_1, y_1)$ with slope $m$ .

Positive Slope: Quantities Rising Together

A positive slope means the vertical variable rises as the horizontal one does. The line climbs from left to right. In physics this is an object moving steadily away from its starting point; the steeper the line, the faster the position changes.

In the graph above, the position rises with time: every second, the object moves 2 meters forward. The graph is a straight line, so the motion is uniform and the velocity constant.

Negative Slope: Quantities Moving Opposite Ways

A negative slope means the vertical variable falls as the horizontal one rises. The line drops from left to right. In physics this is an object returning toward the origin, or moving backward at constant speed.

Here the position falls as time passes. The object starts at $10\text{ m}$ and moves back toward the origin at a steady $2\text{ m/s}$ . The straight line again means constant velocity — now directed the negative way.

Comparing the Two

Both graphs are straight lines, so both show constant velocity. The only difference is the sign of the slope: positive slope, motion in the positive direction; negative slope, motion in the opposite direction. The slope of a position-time graph is the velocity.

Interactive Visualization

Compare the slopes of a linear and a quadratic graph below.

Interactive slope comparison between linear and quadratic functions

In a position-time graph (where $y$ is position and $x$ is time), what physical quantity does the slope of the line represent?

Trigonometric Ratios and Identities

Breaking Down Diagonal Forces

A force is rarely applied along a single axis. A rope pulling a crate makes an angle with the floor, and the pull acts partly forward and partly upward. Trigonometry splits such a force into independent horizontal and vertical components. Without it, angled forces cannot be analysed.

Trigonometry is the study of the relations between the angles and sides of triangles. In a right-angled triangle:

A diagonal force pulling a crate on a flat floor, resolved into horizontal and vertical components with the angle marked.

Six Trigonometric Ratios:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$
$\csc\theta = \frac{1}{\sin\theta}$ (cosecant)
$\sec\theta = \frac{1}{\cos\theta}$ (secant)
$\cot\theta = \frac{1}{\tan\theta}$ (cotangent)

A right triangle with the six trigonometric ratios labeled relative to angle

\theta

The Pythagorean Identities

The opposite, adjacent, and hypotenuse of a right triangle always obey the Pythagorean theorem. Dividing that relation through by each side in turn produces three identities that hold for every angle $\theta$ . They are among the most used relations in physics, letting you pass between $\sin$ , $\cos$ , and $\tan$ at will.

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

$1 + \tan^2\theta = \sec^2\theta$

$1 + \cot^2\theta = \csc^2\theta$

The first identity is the Pythagorean theorem applied to the unit circle. Dividing it through by $\cos^2\theta$ gives the second; dividing it through by $\sin^2\theta$ gives the third.

Standard Trigonometric Values

Certain angles recur so often in physics that their exact sine, cosine, and tangent values are worth committing to memory:

$\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$135^\circ$	$150^\circ$	$180^\circ$
radians	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$	$2\pi/3$	$3\pi/4$	$5\pi/6$	$\pi$
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-\frac{1}{\sqrt{2}}$	$-\frac{\sqrt{3}}{2}$	$-1$
$\tan\theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	Not Defined	$-\sqrt{3}$	$-1$	$-\frac{1}{\sqrt{3}}$	$0$

A force of $10 \text{ N}$ is applied to a block at an angle of $60^\circ$ above the horizontal. What is the magnitude of the horizontal component of this force pulling the block forward? (Given $\cos(60^\circ) = 0.5$ )

Angles: Degrees and Radians

Angles greater than $90^\circ$ matter constantly in physics — a wheel turns through full rotations, not fractions of a right angle. Angles are measured in:

Degrees (°): A full circle = $360^\circ$
Radians (rad): A full circle = $2\pi \text{ rad}$

The radian defined from a circle: the angle theta subtended at the centre equals the arc length s divided by the radius r. — The radian defined from a circle: the angle $\theta$ subtended at the centre equals the arc length $s$ divided by the radius $r$ .

To see what a radian is, take a circle of radius $r$ . The angle $\theta$ (in radians) subtended at the centre by an arc of length $s$ is defined as the ratio of arc length to radius:

$\theta = \frac{s}{r}$

On a unit circle ( $r = 1$ ), the arc length itself is the angle.

A particle moves along a circular path of radius $r = 2 \text{ m}$ . If it travels an arc length of $s = 3 \text{ m}$ , what is the angle subtended at the center of the circle?

The Small Angle Approximation

For very small angles (typically under $10^\circ$ ), the sine of the angle and the angle itself, measured in radians, are nearly equal: $\sin\theta \approx \theta$ . Physics uses this constantly — most famously in deriving the time period of a simple pendulum.

In physics, we frequently use the small angle approximation ( $\sin \theta \approx \theta$ ) to simplify complex equations. For this approximation to be mathematically valid, how MUST the angle $\theta$ be measured?

Angles in All Four Quadrants

In physics, angles greater than $90^\circ$ describe rotations and orientations across the Cartesian plane. By convention, positive angles are measured anti-clockwise from the positive x-axis.

The four quadrants of the unit circle with the ASTC rule: which ratios are positive in each quadrant.

Trigonometric Ratios as Coordinates

Instead of opposite and adjacent sides, the ratios can be defined from the coordinates $(x, y)$ of a point on a circle of radius $r$ :

$\sin\theta = \frac{y}{r}$
$\cos\theta = \frac{x}{r}$
$\tan\theta = \frac{y}{x}$

A point P at distance r from the origin, at angle theta. Its coordinates x and y define the trigonometric ratios of theta. — A point $P$ at distance $r$ from the origin, at angle $\theta$ . Its coordinates $x$ and $y$ define the trigonometric ratios of $\theta$ .

The ASTC Rule (All, Sin, Tan, Cos): The coordinates $x$ and $y$ change sign from quadrant to quadrant, and the ratios change sign with them:

Quadrant I ( $0^\circ$ to $90^\circ$ ): Both x and y are positive. All ratios are positive.
Quadrant II ( $90^\circ$ to $180^\circ$ ): x is negative, y is positive. Only Sine (and cosecant) are positive.
Quadrant III ( $180^\circ$ to $270^\circ$ ): Both x and y are negative. Only Tangent (and cotangent) are positive.
Quadrant IV ( $270^\circ$ to $360^\circ$ ): x is positive, y is negative. Only Cosine (and secant) are positive.

Standard Trigonometric Formulas

The analysis of waves and rotations constantly shifts and combines angles. These are the formulas it runs on.

Phase Shifts and Quadrant Reductions

$\sin(\pi - \theta) = \sin\theta$ and $\cos(\pi - \theta) = -\cos\theta$ (Quadrant II)
$\sin(\pi + \theta) = -\sin\theta$ and $\cos(\pi + \theta) = -\cos\theta$ (Quadrant III)
$\sin(2\pi - \theta) = -\sin\theta$ and $\cos(2\pi - \theta) = \cos\theta$ (Quadrant IV)
$\sin(-\theta) = -\sin\theta$ and $\cos(-\theta) = \cos\theta$ (Even/Odd functions)

Sum, Difference, and Double Angle Formulas

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
$\sin(2\theta) = 2\sin\theta\cos\theta$
$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$

Worked Example 1: Converting Degrees to Radians

Problem: Convert $270^\circ$ into radians.

Solution: Since $\pi \text{ rad} = 180^\circ$ , every degree measure converts on multiplication by $\frac{\pi}{180^\circ}$ :

$270^\circ = \frac{270^\circ}{180^\circ} \times \pi = 1.5\pi \text{ rad}$

Now Solve: Convert $150^\circ$ into radians. (Answer: $\frac{5\pi}{6} \text{ rad}$ )

Worked Example 2: Identifying the Quadrant

Problem: If $\sin\theta < 0$ , $\tan\theta < 0$ , and $\cos\theta > 0$ , in which quadrant does $\theta$ lie?

Solution: Check each condition against the ASTC rule separately, then find the quadrant common to all three:

$\sin\theta < 0$ $\Rightarrow$ 3rd or 4th quadrant
$\tan\theta < 0$ $\Rightarrow$ 2nd or 4th quadrant
$\cos\theta > 0$ $\Rightarrow$ 1st or 4th quadrant

All three conditions are satisfied only in the 4th quadrant.

Now Solve: If $\sin\theta \times \cos\theta > 0$ , in which quadrant(s) does $\theta$ lie? (Answer: 1st or 3rd quadrant)

Worked Example 3: Simplifying a Trigonometric Expression

Problem: Find the value of $\sin 28^\circ \cos 17^\circ + \sin 62^\circ \cos 73^\circ$ .

Solution: Rewrite $62^\circ$ and $73^\circ$ using the co-function identities $\sin(90^\circ - \theta) = \cos\theta$ and $\cos(90^\circ - \theta) = \sin\theta$ , so that both terms share the same angle sum:

$\sin 28^\circ \cos 17^\circ + \sin 62^\circ \cos 73^\circ = \sin 28^\circ \cos 17^\circ + \sin(90^\circ - 28^\circ)\cos(90^\circ - 17^\circ)$

$= \sin 28^\circ \cos 17^\circ + \cos 28^\circ \sin 17^\circ = \sin(28^\circ + 17^\circ) = \sin 45^\circ = \frac{1}{\sqrt{2}}$

Now Solve: Find $\cos 8^\circ \cos 52^\circ - \cos 82^\circ \cos 38^\circ$ . (Answer: $0.5$ )

Worked Example 4: Sum of Two Angles from Their Tangents

Problem: If $\tan A = \frac{1}{2}$ and $\tan B = \frac{1}{3}$ , find $(A+B)$ .

Solution: Apply the tangent sum formula directly:

$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2}\cdot\frac{1}{3}} = \frac{5/6}{5/6} = 1$

Since $\tan(A+B) = 1$ , it follows that $A + B = 45^\circ = \frac{\pi}{4}$ .

Now Solve: Using the same $A$ and $B$ , find $\tan(A-B)$ . (Answer: $\frac{1}{7}$ )

Key Concepts

Numbers, Algebra, and Equations

The number line is a coordinate system. A position is a single real number; the sign carries direction, and the absolute value $|x|$ keeps the size and drops the sign. Displacement is $\Delta x = x_B - x_A$ ; distance is its magnitude $|\Delta x|$ .
Order of operations (BODMAS). Brackets, then exponents, then division and multiplication left to right, then addition and subtraction left to right.
An equation is a balance. Whatever is done to one side — adding, subtracting, multiplying, dividing, squaring, taking a root — must be done to the other. This is the whole method of rearranging formulas: from $F = ma$ , dividing both sides by $m$ gives $a = \frac{F}{m}$ ; from $v = u + at$ , $a = \frac{v - u}{t}$ .
Only like terms combine: $3x + 5x - 2x = 6x$ .
The three variations. Direct: $y = kx$ — doubling $x$ doubles $y$ . Inverse: $y = \frac{k}{x}$ — doubling $x$ halves $y$ (Boyle's law). Inverse square: $y = \frac{k}{x^2}$ — doubling $x$ cuts $y$ to one fourth (gravity, the electric force).

Algebraic Identities Worth Keeping at Hand

$(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$ — note the cross term $2ab$ ; the common error is to drop it.
$a^2 - b^2 = (a + b)(a - b)$ — the difference of squares, the fastest factoring tool in the kit.
$(a \pm b)^3 = a^3 \pm b^3 \pm 3ab(a \pm b)$
$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
$(x + a)(x + b) = x^2 + (a + b)x + ab$

Geometry, Graphs, and Functions

Distance between two points. In a plane, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ ; in three dimensions the formula gains the term $(z_2 - z_1)^2$ . Both are the Pythagorean theorem in coordinates.
Slope is rate of change: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$ . On a position-time graph the slope is the velocity; its sign gives the direction of motion. On a curve, the slope at a point is the slope of the tangent there.
Lines: $y = mx + c$ (slope-intercept) and $y - y_1 = m(x - x_1)$ (point-slope). The intercept $c$ is the initial value.
Quadratics: $y = ax^2 + bx + c$ graphs as a parabola, opening up for $a > 0$ and down for $a < 0$ . The discriminant $b^2 - 4ac$ decides the roots: positive gives two crossings of the x-axis, zero a single touch, negative none. Replacing $x$ by $x - h$ shifts the graph right by $h$ ; adding a constant shifts it up.
Exponentials: $y = a \cdot r^x + c$ . Growth for $r > 1$ , decay towards the asymptote $y = c$ for $0 < r < 1$ . Doubling and half-life are the physical faces of the two cases.

Trigonometry

The ratios. $\sin\theta = \frac{\text{opp}}{\text{hyp}}$ , $\cos\theta = \frac{\text{adj}}{\text{hyp}}$ , $\tan\theta = \frac{\sin\theta}{\cos\theta}$ , with $\csc$ , $\sec$ , $\cot$ their reciprocals. On a circle of radius $r$ : $\sin\theta = \frac{y}{r}$ , $\cos\theta = \frac{x}{r}$ , $\tan\theta = \frac{y}{x}$ .
Component resolution. A force $F$ at angle $\theta$ to the horizontal acts as $F\cos\theta$ horizontally and $F\sin\theta$ vertically. On an incline, gravity splits into $mg\sin\theta$ along the slope and $mg\cos\theta$ into it.
Pythagorean identities: $\sin^2\theta + \cos^2\theta = 1$ , $1 + \tan^2\theta = \sec^2\theta$ , $1 + \cot^2\theta = \csc^2\theta$ .
Radians. $\theta = \frac{s}{r}$ ; a full circle is $2\pi$ rad and $\pi$ rad $= 180^\circ$ . The radian is dimensionless.
Small angles (in radians, under about $10^\circ$ ): $\sin\theta \approx \theta$ — the approximation behind the simple pendulum.
Signs by quadrant (ASTC): all positive in I; only sine in II; only tangent in III; only cosine in IV.
Combining angles: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ , $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ , $\sin 2\theta = 2\sin\theta\cos\theta$ , $\cos 2\theta = \cos^2\theta - \sin^2\theta$ .