Course

O Level Additional Mathematics: Trigonometry, Series and Vectors (4037)

Name: O Level Additional Mathematics: Trigonometry, Series and Vectors (4037)
Price: 1299 BDT

Radians, the six trigonometric functions and their graphs, equations and identities, permutations and combinations, the binomial theorem, progressions and vectors for 4037.

85 lessons14h 37mEnglish

What you'll learn

Use radians to find arc lengths, sector areas, segments and compound shapes
Use the six trigonometric functions for angles of any size, with exact values
Draw and use the graphs of y = a sin bx + c, y = a cos bx + c, y = a tan bx + c and their modulus
Solve trigonometric equations in a given domain and prove trigonometric identities
Count arrangements and selections with n!, nPr and nCr, and decide when order matters
Expand (a + b)ⁿ with the binomial theorem and find any term with the general term
Use arithmetic and geometric progressions, including the sum to infinity
Use vectors, position and unit vectors, vector geometry and constant velocity problems

Requirements

Cambridge O Level Mathematics 4024 or equivalent, including right-angled trigonometry, indices and surds
Solving quadratic equations by factorising and by the formula

Course content

85 lessons · 14h 37m

9.1 · Circular measure

Measure angles in radians, change between degrees and radians, and find arc lengths, sector areas, segments and compound shapes.

0/9

Radians

One radian is the angle where the arc is as long as the radius. A full turn is 2π radians.

Preview10m 2.

Converting between degrees and radians

π radians is 180°, so multiply by π/180 to get radians and by 180/π to get degrees.

Preview10m

Length of an arc

With the angle θ in radians, the arc length is s = rθ. The perimeter of a sector adds two radii.

11m

Area of a sector

A sector is the fraction θ/2π of the whole circle, so its area is ½r²θ with θ in radians.

11m

Chords and segments

A segment is a sector with its triangle taken away: area ½r²θ − ½r² sin θ.

12m

Problems involving compound shapes

Split the shape into sectors, triangles and rectangles. Add the pieces you keep and subtract the holes.

13m

Circular measure: practice

A timed drill of mixed questions on radians, arcs, sectors and segments, each with a worked answer.

Circular measure: exam practice

Past-paper style questions for Paper 1 and Paper 2, with full working and mark schemes.

15m

Circular measure: summary

Every formula, key fact and common mistake from the chapter on one screen.

1.4, 10.1-10.3 · Trigonometry: functions and graphs

The six trigonometric functions for angles of any size, then the graphs of y = a sin bx + c, a cos bx + c and a tan bx + c, their amplitude and period, and |f(x)|.

0/12

The six trigonometric functions

Secant, cosecant and cotangent are 1 over cosine, sine and tangent. Exact values come from two special triangles.

Preview11m 2.

Angles of any magnitude

Angles turn anticlockwise from the positive x-axis. Negative angles turn clockwise, and angles can pass 360°.

Preview10m

Trigonometric ratios of general angles

Find the basic acute angle, then the sign from the quadrant. sin 210° = −sin 30° = −½.

12m

Graphs of sine, cosine and tangent

A point going round a circle draws the sine and cosine waves. The tangent graph has asymptotes where cos x = 0.

11m

Amplitude and period

In y = a sin bx, the amplitude is a and the period is 360° ÷ b. Compare y = sin x with y = 3 sin 2x.

11m

Graphs of y = a sin bx + c and y = a cos bx + c

Adding c moves the whole wave up. The maximum is c + a and the minimum is c − a.

12m

Graphs of y = a tan bx + c

The period of tan bx is 180° ÷ b. Label every asymptote with its x-coordinate.

11m

Graphs of y = |f(x)|, where f(x) is a trigonometric function

Keep the part above the x-axis and reflect the part below it upwards.

11m

Using trigonometric graphs

Read a, b and c off a graph, and count the solutions of an equation by drawing a straight line.

12m

10.

Trigonometric functions and graphs: practice

A timed drill of mixed questions on signs, exact values, amplitude, period and graphs.

11.

Trigonometric functions and graphs: exam practice

Past-paper style questions on exact values and sketching graphs, with full working and mark schemes.

15m

12.

Trigonometric functions and graphs: summary

The six functions, the quadrant signs and the graph facts on one screen.

10.4-10.6 · Trigonometry: equations and identities

Solve trigonometric equations in a given domain, use sin²A + cos²A = 1, sec²A = 1 + tan²A and cosec²A = 1 + cot²A, and prove identities.

0/13

Trigonometric equations

Find the basic angle, use the quadrant signs, and list every solution in the given domain.

Preview12m 2.

Equations with secant, cosecant and cotangent

Turn sec x = 2 into cos x = ½ first. Then solve as before.

Preview10m

Equations with multiple angles

For sin 2x = k, stretch the domain for 2x first, solve for 2x, then halve every answer.

12m

Equations with compound angles, such as 2x − π/3

Treat the whole bracket as one angle. Find its domain, solve, then undo the steps to reach x.

12m

Trigonometric identities: tan A = sin A / cos A and sin²A + cos²A = 1

A point on a circle of radius 1 is (cos A, sin A). Pythagoras gives sin²A + cos²A = 1.

11m

Trigonometric identities: sec²A = 1 + tan²A and cosec²A = 1 + cot²A

Divide sin²A + cos²A = 1 by cos²A or by sin²A to get the other two identities.

10m

Further trigonometric equations: quadratics in one function

Let s = sin x, solve the quadratic, then reject any value outside −1 to 1.

12m

Further trigonometric equations: using identities

Use an identity to write the equation in ONE function, such as tan x, then solve.

13m

Proving trigonometric identities

Start from the harder side, change everything into sin and cos, and use one identity at a time.

12m

10.

Further trigonometric identities

Add fractions over a common denominator, then look for 1 − cos²A or sin²A + cos²A.

13m

11.

Trigonometric equations and identities: practice

A timed drill of mixed equations and identity steps, each with a worked answer.

12.

Trigonometric equations and identities: exam practice

Past-paper style "solve" and "show that" questions with full working and mark schemes.

15m

13.

Trigonometric equations and identities: summary

The three identities, the steps for solving and proving, and the common mistakes on one screen.

11.1-11.3 · Permutations and combinations

Use n!, nPr and nCr to count arrangements and selections, with conditions, and decide when order matters.

0/12

Factorial notation

n! = n × (n − 1) × … × 1 counts the ways to line up n different things, and 0! = 1.

Preview10m 2.

Arrangements

Fill the places one at a time and multiply the number of choices for each place.

Preview11m

Arrangements with conditions

Fill the fixed places first. Glue together people who must sit together, and subtract for "not together".

12m

Permutations

Arranging r of n different things gives nPr = n!/(n − r)!, the first r factors of n!.

11m

Permutations with conditions

For codes that must be even or greater than 5000, fill the restricted place first.

12m

Combinations

Every group of r can be ordered in r! ways, so nCr = nPr ÷ r! = n!/(r!(n − r)!).

12m

Combinations with conditions

Put "must" people in first. Split "at least" into cases, multiply inside each case and add the cases.

13m

Permutations or combinations?

Swap two chosen items. If the result changes, order matters: use nPr. If not, use nCr.

11m

Algebraic problems: finding n

Write nC2 = 45 as n(n − 1)/2 = 45, solve the quadratic, and reject the negative root.

11m

10.

Permutations and combinations: practice

A timed drill: decide order or no order, then count. Every answer has a worked reason.

11.

Permutations and combinations: exam practice

Past-paper style arrangement and selection questions with full working and mark schemes.

15m

12.

Permutations and combinations: summary

n!, nPr, nCr, the order test and the condition tricks on one screen.

12.1-12.2 · Series: the binomial theorem

Expand (a + b)ⁿ with Pascal's triangle and the binomial theorem, and use the general term to find one term or an unknown constant.

0/8

Pascal's triangle

Each number is the sum of the two above it. Row n gives the coefficients of (a + b)ⁿ.

Preview10m 2.

Expanding (a + b)ⁿ using Pascal's triangle

Powers of a go down, powers of b go up. Put terms like (−3x) in brackets before you raise them.

Preview12m

The binomial theorem

The coefficients in row n are nC0, nC1, …, nCn, so large powers need no triangle.

12m

The general term

The term with bʳ is nCr aⁿ⁻ʳ bʳ. Set the power of x equal to what you want and solve for r.

12m

Using the binomial theorem: coefficients and unknown constants

Find a coefficient in (1 + 2x)(1 − x)⁶, or find k when a coefficient is given.

13m

The binomial theorem: practice

A timed drill of expansions, single terms and unknown constants, each with a worked answer.

12m

The binomial theorem: exam practice

Past-paper style binomial questions with full working and mark schemes.

15m

The binomial theorem: summary

Pascal's triangle, the binomial formula and the general term on one screen.

12.3-12.5 · Series: arithmetic and geometric progressions

Recognise arithmetic and geometric progressions, use their nth term and sum formulas, and find the sum to infinity when it exists.

0/9

Arithmetic progressions

Add the same common difference d each time. The nth term is a + (n − 1)d.

Preview11m 2.

The sum of an arithmetic progression

Write the sum forwards and backwards. Each pair adds to a + l, so Sₙ = ½n(a + l) = ½n(2a + (n − 1)d).

Preview12m

Geometric progressions

Multiply by the same common ratio r each time. The nth term is arⁿ⁻¹.

11m

The sum of a geometric progression

Multiply the sum by r and subtract. Almost every term cancels, leaving Sₙ = a(1 − rⁿ)/(1 − r).

12m

The sum to infinity of a geometric progression

When −1 < r < 1, rⁿ shrinks to 0 and the sum settles at a/(1 − r). Otherwise there is no sum to infinity.

12m

Further arithmetic and geometric progressions

Check differences and ratios before you pick a formula, then solve problems that use both kinds.

13m

Arithmetic and geometric progressions: practice

A timed drill: name the progression, then find a term, a sum or a sum to infinity.

12m

Arithmetic and geometric progressions: exam practice

Past-paper style progression questions, some in context, with full working and mark schemes.

15m

Arithmetic and geometric progressions: summary

The nth term, sum and sum to infinity formulas side by side, with the common mistakes.

13.1-13.4 · Vectors in two dimensions

Write vectors in every notation, find magnitudes and unit vectors, use position vectors and vector geometry, and solve constant velocity problems.

0/15

Vector notation

One move can be written as a column, as AB with an arrow, as a bold letter, or as 3i + 4j.

Preview10m 2.

The magnitude of a vector

The magnitude is the length of the arrow: |xi + yj| = √(x² + y²) by Pythagoras.

Preview10m

Adding and subtracting vectors

Put the arrows head to tail. In components, add or subtract the i parts and the j parts.

11m

Multiplying a vector by a scalar

kv points the same way k times as long. A negative k turns it round. Parallel vectors are multiples.

10m

Position vectors

The position vector of A goes from the origin O to A. The vector from A to B is b − a.

11m

Unit vectors

Divide a vector by its magnitude to get a unit vector: length 1, same direction.

10m

Equating like vectors

If two vectors are equal, their i parts are equal and their j parts are equal. Solve for the unknowns.

11m

Vector geometry

Write each route in the same two vectors. A multiple shows lines are parallel, or points lie on one line.

13m

Composing velocities

Add velocity vectors to get the resultant velocity. Its magnitude is the actual speed.

12m

10.

Resolving velocities

Velocity = speed × unit vector in the direction of travel. A speed at an angle splits into i and j parts.

11m

11.

Constant velocity problems: position after time t

Start position plus t times the velocity: r = r₀ + tv gives the position at any time.

12m

12.

Constant velocity problems: collisions

Two objects collide if they are at the same place at the same time. Equate the i parts and the j parts.

13m

13.

Vectors in two dimensions: practice

A timed drill of magnitudes, unit vectors, routes and velocities, each with a worked answer.

14.

Vectors in two dimensions: exam practice

Past-paper style vector geometry and velocity questions with full working and mark schemes.

15m

15.

Vectors in two dimensions: summary

Notation, magnitude, unit vectors, routes and r = r₀ + tv on one screen.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.