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\n"},{"id":106,"name":"Quadratics","slug":"quadratics","topic":2,"subjects":["AAHL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"-f2Vrii1MY8","concepts":[{"id":404,"name":"Quadratics and parabolas","slug":"quadratics-and-parabolas","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770853908080-quadratics.png?alt=media&token=e4ba899b-9802-4e8e-9c73-a720fbeb521e","type":"image"},"skills":[806,805,406,407,800,411,408,405,845,801,802],"description":"Quadratic expressions \$ax^2+bx+c\$, their parabola graphs, the axis of symmetry, vertex, and intercetps.","hasLesson":true},{"id":223,"name":"Applications of Quadratics","slug":"applications-of-quadratics","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770854104447-quadratic%20applications.png?alt=media&token=fe504cd1-e3c5-4205-9363-26479955bca3","type":"image"},"skills":[766,924],"description":"

Evaluating the discriminant, sum and product of roots, quadratic inequalities, quadratics of other forms.

\n","hasLesson":true}],"description":"

Quadratic expressions and functions in standard, factored and vertex form, including graphing parabolas, roots, axis of symmetry and vertex, factoring, completing the square, quadratic formula, discriminant, inequalities, calculator methods and simple optimization.

\n"},{"id":99,"name":"Exponents & Logarithms","slug":"exponents-logarithms","topic":1,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"Y_x-KthdQmc","concepts":[{"id":168,"name":"Exponentiation and integer exponent laws","slug":"exponentiation-and-integer-exponent-laws","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030280792-Artboard%201.png?alt=media&token=6be9f1a7-c875-4ed1-bf66-e1be44cf57e4","type":"image"},"skills":[620,621,614,618,616,622,619,613,844],"description":"

Exponential notation and algebra, including laws for zero and negative exponents, multiplying and dividing powers with the same base, products and quotients with the same exponent, powers of powers, equating exponents in exponential equations, and rationalizing denominators when needed.

\n","hasLesson":true},{"id":879,"name":"Radicals and roots","slug":"radicals-and-roots","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030066026-Artboard%203.png?alt=media&token=8e47da95-01c8-40c1-8577-c825a8f16f5f","type":"image"},"skills":[1072,886,888,889,892,893,895,897],"description":"

Square roots and nth roots, including \\sqrt[n]{a}=a^{\\frac1n}, rational exponents a^{\\frac{m}{n}}, simplifying radicals, roots of negative numbers, and rationalizing denominators using roots or conjugates.

\n","hasLesson":true},{"id":171,"name":"Logarithm algebra","slug":"logarithm-algebra","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030072565-Artboard%202.png?alt=media&token=7a029ca5-f0a3-452e-8694-6ab671866da3","type":"image"},"skills":[624,625,360,626,361,1066,628,627,362],"description":"

Logarithm algebra including the definition \\log_a b=x \\Leftrightarrow a^x=b, base 10 and natural logs, evaluating logs exactly or with technology, and using the rules \\log_a x+\\log_a y=\\log_a(xy), \\log_a x-\\log_a y=\\log_a(x/y), \\log_a(x^m)=m\\log_a x, change of base, and logarithms to solve exponential equations.

\n","hasLesson":true},{"id":170,"name":"Exponential functions","slug":"exponential-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030057079-Artboard%204.png?alt=media&token=eabb4444-acbe-474e-8671-d9c3c8b4b523","type":"image"},"skills":[493,494,797,904,901,902],"description":"

Exponential functions f(x)=a^x and logarithmic functions f(x)=\\log_a x, their domains and ranges, inverse relationship and symmetry in y=x, and graphing/making sense of exponential growth and decay models such as ca^x+k or Ab^t+c.

\n","hasLesson":true}],"description":"

Exponents, roots and rational powers, logarithm laws and change of base, and exponential and logarithmic functions and equations.

\n"},{"id":104,"name":"Function Theory","slug":"function-theory","topic":2,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"5-tzjROivOo","concepts":[{"id":252,"name":"Functions and their properties","slug":"functions-and-their-properties","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770684953385-function%20machine.png?alt=media&token=c59ac711-975a-4c73-ae39-f10fda053427","type":"image"},"skills":[384,1073,1074,1075,1085],"description":"The concept of a function as a mathematical machine, the notation f\\left(x\\right)=\\ldots, and function tables / diagrams.","hasLesson":true},{"id":258,"name":"Function Graphs","slug":"function-graphs","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770685518993-function%20graphs.png?alt=media&token=d16df97f-db93-49ec-a84f-bd64de28e3ec","type":"image"},"skills":[415,847,849,260,505,373,391,375,374,376,1035,907],"description":"

Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima

","hasLesson":true},{"id":968,"name":"Domain and range","slug":"domain-and-range","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770684976612-Artboard%2034.png?alt=media&token=4121dfaf-2525-499a-9513-5d0fc2785d56","type":"image"},"skills":[758,759,848,762],"description":"

The domain of a function is the set of possible inputs, and the range is the set of possible outputs.

","hasLesson":true},{"id":253,"name":"Function Composition","slug":"function-composition","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770854427967-function%20composition.png?alt=media&token=81948aeb-8bb9-4784-8de2-4a1944a38553","type":"image"},"skills":[763],"description":"

Composing functions like f\\circ g

","hasLesson":true},{"id":970,"name":"Inverse Functions","slug":"inverse-functions","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1770854455145-inverse%20functions.png?alt=media&token=539c397b-5ca2-4e26-9810-ce37225be884","type":"image"},"skills":[386,378,379,416,377,380],"description":"The concept of an inverse function, its graph as a reflection in the line y=x, finding the inverse of a specific value, and domain and range of inverse functions. When inverse functions exist, and how to find them when they do.","hasLesson":true}],"description":"

Function notation and representations, including evaluating, graphing and modeling functions, domain and range, composition and inverse functions, and using technology to analyze graphs, intercepts, extrema and asymptotes.

\n"},{"id":865,"name":"Rounding & Error","slug":"rounding-error","topic":1,"subjects":["AISL","AASL","AIHL","AAHL"],"isVideoReady":false,"videoId":null,"concepts":[{"id":866,"name":"Rounding and estimation","slug":"rounding-and-estimation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030041198-Artboard%205.png?alt=media&token=25922b18-0ebb-4291-af0d-e8584c14c0e9","type":"image"},"skills":[869,877],"description":"

Rounding, significant figures, decimal places

","hasLesson":true},{"id":868,"name":"Scientific Notation","slug":"scientific-notation","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1773515918578-Artboard%206.png?alt=media&token=69a03129-81d4-4eeb-a181-bef451d1b1cd","type":"image"},"skills":[882,885,887],"description":"

Converting numbers to and from scientific notation \\left(a\\times10^{k}\\right)

","hasLesson":true}],"description":"

Rounding, significant figures, and scientific notation.

"},{"id":30,"name":"Sequences & Series","slug":"sequences-series","topic":1,"subjects":["AAHL","AIHL","AISL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"DI2cudLIWEk","concepts":[{"id":27,"name":"Arithmetic Sequences","slug":"arithmetic-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029981178-Artboard%207.png?alt=media&token=8a340859-acc8-458a-a5be-e962cb927b61","type":"image"},"skills":[602,603],"description":"

Introduction to the definition, properties, and general term of an arithmetic sequence

","hasLesson":true},{"id":28,"name":"Geometric Sequences","slug":"geometric-sequences","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030011943-Artboard%208.png?alt=media&token=96b137e1-fd1c-43b5-9749-9222bc9e9393","type":"image"},"skills":[604,40],"description":"

Introduction to definition, identification, and general term of geometric sequences

","hasLesson":true},{"id":31,"name":"Arithmetic Series","slug":"arithmetic-series","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757030004817-Artboard%209.png?alt=media&token=4f2e2c52-6ec5-4ca1-b258-67a93d257f07","type":"image"},"skills":[607,46],"description":"

Definition of series, formula for arithmetic series

","hasLesson":true},{"id":860,"name":"Summation notation","slug":"summation-notation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029991958-Artboard%2010.png?alt=media&token=dcfdc9b6-9e57-47bc-b6bc-9b4088d76eb5","type":"image"},"skills":[47,837,861,862,863,905],"description":"

Summation notation, properties of sums

","hasLesson":true},{"id":859,"name":"Geometric Series","slug":"geometric-series","isPaywalled":true,"isPublic":true,"thumbnail":{"type":"graph","graph":{"data":"$48","mode":"geometry"}},"skills":[606,608,51],"description":"

Finite geometric series with sum S_{n}=\\frac{u_1(1-r^n)}{1-r}, and infinite geometric series convergence when |r|<1 so that S_\\infty=\\frac{u_1}{1-r}.

\n","hasLesson":true},{"id":906,"name":"Sequence Mode (Calculator)","slug":"sequence-mode-calculator","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029904498-Artboard%2011.png?alt=media&token=542aa577-0f10-43f9-b206-99a890877c96","type":"image"},"skills":[],"description":"

Working with sequences on a calculator, computations and applications of a calculator

","hasLesson":true},{"id":870,"name":"Compound interest, depreciation and inflation","slug":"compound-interest-depreciation-and-inflation","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029508658-Artboard%2012.png?alt=media&token=fb6f46c3-0587-4c9c-83b6-2dd08aad7ca4","type":"image"},"skills":[872,871,883,896,873],"description":"

Compound growth, depreciation, interest, inflation, real value

","hasLesson":true}],"description":"Arithmetic sequences u_{n}=u_1+\\left(n-1\\right)d and geometric sequences v_{n}=v_1r^{n-1}, as well as their series and the \\Sigma notation for summation."},{"id":100,"name":"Counting & Binomials","slug":"counting-binomials","topic":1,"subjects":["AAHL","AASL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"w08O6aqQegU","concepts":[{"id":246,"name":"Pascal's Triangle and nCr","slug":"pascals-triangle-and-ncr","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029889677-Artboard%2013.png?alt=media&token=8abbdcf6-30e5-4df2-a898-50033c4a5db7","type":"image"},"skills":[631,630,629,1025],"description":"

Pascal's triangle and binomial coefficients, using factorials and ^{n}C_{r}=\\frac{n!}{r!(n-r)!} to find combinations and the coefficients in binomial expansions.

\n","hasLesson":true},{"id":248,"name":"Binomial Theorem","slug":"binomial-theorem","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029881955-Artboard%2014.png?alt=media&token=1442315c-0b38-4a37-a959-022b43646b65","type":"image"},"skills":[249,632],"description":"

Expansion of \\left(a+b\\right)^n using binomial coefficients, including the general term \\binom{n}{r}a^{n-r}b^r and finding the coefficient of a specified term.

\n","hasLesson":true}],"description":"

Factorials, combinations and Pascal's triangle, and using the binomial theorem to expand \\left(a+b\\right)^n and find specific coefficients.

\n"},{"id":102,"name":"Proof and Reasoning","slug":"proof-and-reasoning","topic":1,"subjects":["AAHL","AASL"],"isVideoReady":true,"videoId":"syZ6AKBabRA","concepts":[{"id":240,"name":"Proof by deduction","slug":"proof-by-deduction","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029849287-Artboard%2017.png?alt=media&token=312ca3d8-5599-4f64-8439-464901c0de57","type":"image"},"skills":[652,650],"description":"

Strict logical proofs by deduction, including parity arguments with even and odd integers written as n=2k and n=2k+1, and direct proofs that two expressions are equivalent by showing \\text{LHS}\\equiv\\text{RHS} for all variables.

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Simple deductive proofs to show LHS\\equiv RHS.

"},{"id":108,"name":"Transformations & asymptotes","slug":"transformations-asymptotes","topic":2,"subjects":["AAHL","AASL","AIHL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"xr5EGY1f9Ps","concepts":[{"id":263,"name":"Linear Transformations","slug":"linear-transformations","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029188684-Artboard%2038.png?alt=media&token=eab72306-0d65-4d6a-882d-85b179795b8d","type":"image"},"skills":[11,769,770,771,772,773,774],"description":"

Linear transformations of graphs and points: translations by a vector \\begin{pmatrix}a\\\\ b\\end{pmatrix}, vertical and horizontal stretches y=af(x) and y=f(qx), vertical and horizontal translations y=f(x)+b and y=f(x-a), and reflections in the x- and y-axes.

\n","hasLesson":true},{"id":255,"name":"Rational functions","slug":"rational-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029160193-Artboard%2039.png?alt=media&token=cced4769-0d16-429a-8ae4-bd3a62d893ca","type":"image"},"skills":[808,364],"description":"

The reciprocal function f\\left(x\\right)=\\frac{1}{x}, its undefined value at x=0, self-inverse property, and the vertical and horizontal asymptotes of linear rational functions f\\left(x\\right)=\\frac{ax+b}{cx+d}.

\n","hasLesson":true}],"description":"

Translations, reflections and stretches of graphs, including reciprocal and linear rational functions with vertical and horizontal asymptotes.

\n"},{"id":356,"name":"2D & 3D Geometry","slug":"2d-geometry","topic":3,"subjects":["AAHL","AASL","AIHL","AISL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"GnG2jU8VpUg","concepts":[{"id":82,"name":"Right-angled triangles","slug":"right-angled-triangles","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029099320-Artboard%2046.png?alt=media&token=803487ed-7a2d-4c58-88e5-31cba1785454","type":"image"},"skills":[87,36,56,716,83,717],"description":"

Pythagoras's theorem, SOHCAHTOA, finding side length from angles

","hasLesson":true},{"id":37,"name":"Non-right-angled triangles","slug":"non-right-angled-triangles","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757029088911-Artboard%2047.png?alt=media&token=80493296-567f-4f51-b0b3-4c328ab90285","type":"image"},"skills":[88,73,74,718],"description":"

Area of non-right-angled triangles using A=\\frac12ab\\sin C, the sine rule \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=\\frac{c}{\\sin C}, the cosine rule c^2=a^2+b^2-2ab\\cos C, and the ambiguous case when solving triangles.

\n","hasLesson":true},{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,68,4,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\pi r^2.

\n","hasLesson":true},{"id":722,"name":"3D volume and surface area","slug":"3d-volume-and-surface-area","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028955870-Artboard%2050.png?alt=media&token=3f2ceb70-0712-4f69-81cc-c257607a3013","type":"image"},"skills":[1080,1094,723,724,725,834,835],"description":"

Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids

","hasLesson":true},{"id":89,"name":"Angles of elevation and depression","slug":"angles-of-elevation-and-depression","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028981407-Artboard%2048.png?alt=media&token=5b9c00d8-8c4e-40c8-afef-26030c38db55","type":"image"},"skills":[90],"description":"Angles measured between a horizontal line and some elevated or depressed object.","hasLesson":true}],"description":"

2D and 3D geometry including circle measures and sectors, right-triangle and non-right-triangle trigonometry, bearings and angles of elevation/depression, 3D coordinates and angles in space, and surface area and volume of common solids.

\n"},{"id":58,"name":"Trig equations & identities","slug":"trig-equations-identities","topic":3,"subjects":["AAHL","AASL","AIHL","ecolint-y11-hl"],"isVideoReady":true,"videoId":"M87nP0Y-Pds","concepts":[{"id":65,"name":"Radians, arcs and sectors","slug":"radians-arcs-and-sectors","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028966537-Artboard%2049.png?alt=media&token=21e1f320-b306-4587-8651-9991150cbd34","type":"image"},"skills":[492,66,720,68,4,14],"description":"

Radian measure, converting between radians and degrees, and calculating arc length and sector area using l=r\\theta, A=\\frac12\\theta r^2, C=2\\pi r, and A=\\pi r^2.

\n","hasLesson":true},{"id":166,"name":"The Unit Circle","slug":"the-unit-circle","isPaywalled":false,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028896524-Artboard%2053.png?alt=media&token=6f0c6598-c2ab-4090-99f1-8e7ccf54e5df","type":"image"},"skills":[69,70,729,730,728,732,731,734,389,61],"description":"

The unit circle: using \\cos\\theta and \\sin\\theta as the x- and y-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as \\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta}, and \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":735,"name":"Sinusoidal functions","slug":"sinusoidal-functions","isPaywalled":false,"isPublic":true,"thumbnail":{"type":"graph","graph":{"data":"$49","mode":"geometry"}},"skills":[736,76,14,75],"description":"

Radian measure, the sine and cosine functions with domain x\\in\\R and range (-1,1), the tangent function \\tan x=\\frac{\\sin x}{\\cos x} with its asymptotes and roots, and sinusoidal functions of the form a\\sin\\left(b\\left(x+c\\right)\\right)+d or a\\cos\\left(b\\left(x+c\\right)\\right)+d.

\n","hasLesson":true},{"id":949,"name":"Tangent and reciprocal trig functions","slug":"tangent-and-reciprocal-trig-functions","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028852382-Artboard%2054.png?alt=media&token=c153b834-4d14-48e7-b00c-ce7ed61b764b","type":"image"},"skills":[],"description":"

Reciprocal trigonometric functions: secant \\sec x = \\frac{1}{\\cos x}, cosecant \\cosec x = \\frac{1}{\\sin x}, and cotangent \\cot x = \\frac{1}{\\tan x}, including their basic values and graph behaviour where defined.

\n","hasLesson":true},{"id":396,"name":"Trigonometric equations","slug":"trigonometric-equations","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028815933-Artboard%2055.png?alt=media&token=283b491c-074b-4773-bc66-4c839b2c8cb5","type":"image"},"skills":[746,1021,744,745,747],"description":"

Solving trigonometric equations algebraically in a specified domain, including \\sin\\theta=a, \\cos\\theta=a, \\tan\\theta=a, equations with arguments of the form ax+b, and trigonometric quadratics using identities such as \\sin^2\\theta+\\cos^2\\theta=1.

\n","hasLesson":true},{"id":393,"name":"Pythagorean and double-angle identities","slug":"pythagorean-and-double-angle-identities","isPaywalled":true,"isPublic":true,"thumbnail":{"url":"https://firebasestorage.googleapis.com/v0/b/perplex-learning.appspot.com/o/thumbnails%2F1757028794001-Artboard%2057.png?alt=media&token=6c43949e-408e-48a5-aa5d-2ee8a9e57c20","type":"image"},"skills":[61,739,740],"description":"

Trigonometric identities including \\sin^2\\theta+\\cos^2\\theta=1, \\sin2\\theta=2\\sin\\theta\\cos\\theta, and the equivalent forms of \\cos2\\theta: \\cos^2\\theta-\\sin^2\\theta, 2\\cos^2\\theta-1, and 1-2\\sin^2\\theta.

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Unit circle trigonometry, key values, quadrant/sign rules, symmetry and periodicity, double angle and Pythagorean identities, and solving basic trigonometric equations and trigonometric quadratics in a given domain.

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The basic idea of a limit \\lim_{x\\to a}f(x) from tables and graphs, slope as a limit, rate of change and gradient, derivatives of x^n, and linearity for sums and scalar multiples.

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Derivatives of x^{r},e^{x},\\ln x,\\sin x and \\cos x. The chain, product and quotient rules.

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Definitions of tangent and normal, finding tangent and normal lines to a function

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Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation

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Second derivatives f'', using the sign of f'' to determine concavity, locate inflexion points where f''(x)=0 and changes sign, and classify stationary points with the second derivative test.

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Limits and rates of change, differentiation rules for polynomial, exponential, logarithmic and trigonometric functions, and applications to tangents, normals, stationary points, concavity and optimisation.

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Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills

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Antiderivatives of polynomials, trigonometric, inverse trigonometric, exponential, and logarithmic functions

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Integration of composite functions using substitution, including integrals of the form f(ax+b), and handling definite integrals by changing the bounds or substituting back.

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Displacement and distance, velocity and acceleration, speed as |v|, average velocity and acceleration as \\Delta s/\\Delta t and \\Delta v/\\Delta t, and the relations v=\\mathrm{d}s/\\mathrm{d}t, a=\\mathrm{d}v/\\mathrm{d}t=\\mathrm{d}^2s/\\mathrm{d}t^2, \\int_{t_1}^{t_2} v(t)\\,\\mathrm{d}t for change in displacement, and \\int_{t_1}^{t_2} |v(t)|\\,\\mathrm{d}t for distance.

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Integration as reverse differentiation, including basic anti-derivatives, definite integrals, area under and between curves, substitution, and applications to displacement and distance.

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Theoretical and experimental probability, complementary events, expected number of outcomes, sample space

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Venn diagrams, intersection and union of events, conditional probability, independent events

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Tree diagrams, selection with/without replacement, dependence

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Theoretical vs experimental probability, combined events, independence vs dependence.

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Understand the basics of population, data collection, and sampling.

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Learn different ways to measure the \"center,\" or typical values, of a set of data: mean, median, and mode.

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Learn the concept of dispersion, range, IQR, outliers, and box and whisker plots.

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Learn about standard deviation and variance, which we use to measure how tightly or loosely clustered the data is around the mean.

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Summarizing and interpreting data using measures of centre and spread, frequency tables and graphs, sampling methods, quartiles, IQR, box plots, cumulative frequency, and outlier analysis.

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Scatter diagrams and lines of best fit y=ax+b, by eye or with GDC, the Pearson correlation coefficient r, and making predictions with awareness of limitations.

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Correlation and linear regression for paired data, including best fit lines, Pearson's r, predictions from regression models, and the limits of extrapolation.

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Concept of the binomial distribution, binomial PDF and CDF, expectation and variance of binomial distribution

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The normal (bell-shaped) distribution X\\sim\\mathrm{N}(\\mu,\\sigma^2), its symmetry about \\mu, z-values z=\\frac{x-\\mu}{\\sigma} and standardization to Z\\sim\\mathrm{N}(0,1), the empirical rule, and calculator-based normal and inverse normal probability calculations.

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Discrete random variables and probability distributions, including expected value and fair games, binomial models and calculator probabilities, and the normal distribution with z-values and standardization.

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