• Estimating and Approximation

    We could use Pythagoras theorem to find the exact length of the hypotenuse, the longest side in the right angled triangle above. To check the calculation we could find Example: Estimate We work out Example Estimate We work out

    https://astarmathsandphysics.com/gcse-maths-notes/588-estimating-and-approximation.html

  • Circular Measure

    is just the area of the circle times the angle subtended by the arc forming the sector divided by The area formed by the triangle formed by two radii and a chord is The area formed by the difference between a sector and a segment is given by subtracting...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1119-circular-measure.html

  • Proof of the Cosine Rule

    With a triangle labelled as below the Cosine Rule states to prove the rule, drop a perpendicular from to the side Pythagoras Theorem gives Equating these expressions gives hence From the above diagram, so

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1134-proof-of-the-cosine-rule.html

  • Proof of the Sine Rule

    The Sine Rule states that for the triangle labelled as below, To prove it, start by dropping a perpendicular from a vertex to cut the opposite side at right angles, say from to b. Label this perpendicular Then so (1) Now draw a line from to cut the...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1135-proof-of-the-sine-rule.html

  • Symbols and Notation

    line segment with endpoints and or the length of the line containing points and the angle at or the angle between and the triangle with vertices the vector the vector represented by the line segment from to the unit vectors in the directions of the and...

    https://astarmathsandphysics.com/?id=1335:symbols-and-notation&catid=97

  • Cauchy Sequences

    sequence. Proof: Suppose converges to Choose then and there is a positive integer such that implies and hence by the triangle inequality hence is Cauchy. If a sequence is convergent, it must be Cauchy. If it is not Cauchy, it is not convergent. To prove...

    https://astarmathsandphysics.com/university-maths-notes/analysis/1736-cauchy-sequences.html

  • Proof That Any Vector Field Represented by a Symmetric Jacobian Matrix is Conservative

    x_{q_2}e_{q_2}... x_{q_m}e_{q_m}\] Each path segment occur once in each of the two paths since each is the side of a triangle, opposite sides being the same length, so each path can be transformed into the other by a sequence of interchanging of...

    https://astarmathsandphysics.com/university-maths-notes/vector-calculus/4071-proof-that-any-vector-field-represented-by-a-symmetric-jacobian-matrix-is-conservative.html

  • Proof That the Angle Subtended by a Chord or Arc at the Center of a Circle is Twice the Angle Subtended at the Circumference

    Triangle is isosceles since is isosceles similarly. We can labels the angles as below. Then and as below. Then the remaining angles as below (since is a straight line, so and similarly for BOC) Then

    https://astarmathsandphysics.com/gcse-maths-notes/636-proof-that-the-angle-subtended-by-a-chord-or-arc-at-the-center-of-a-circle-is-twice-the-angle-subtended-at-the-circumference.html

  • Centres of Mass, Summation Method

    For a uniform regular body with symmetry eg a cube, flat triangle, sphere, the centre of mass is in the middle. Unfortunately, most bodies are not regular or uniform. To find the centre of mass of an arbitrary body we must use one of the formulae, or...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3634-centres-of-mass-summation-method.html

  • Lamina Suspended at Angle to Horizontal

    resolving vertically and taking moments about a point where one of the strings is attached to the lamina. The equilateral triangle of mass in the form of a lamina of mass below is attached to two strings, one a quarter of a side from a vertex and the...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3640-lamina-suspended-at-angle-to-horizontal.html

  • Proof of Law of Quadratic Reciprocity

    \rightarrow p | a{p}a\] and \[q | b\] , which forces \[(a,b)\] outside the rectangle. Labelling the regions in the triangle above and below the diagonal from the origin to \[(a,b)\] . For \[0 \lt k \lt \frac{p}{2}\] the number of lattice points in B...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5173-proof-of-law-of-quadratic-reciprocity.html

  • When is a Rational Fraction a Convergent of an Infinite Continued Fraction

    \[q_r \| x - \frac{p_r}{q_r} \| \lt \frac{1}{2b} \rightarrow \| x- \frac{p_r}{q_r} \| \lt \frac{1}{2q_rb}\] . The Triangle Inequality gives \[\| \frac{a}{b} - \frac{p_r}{q_r} \| \le \| x - \frac{p_r}{q_r} \| + \| \frac{a}{b} - x \| \lt \frac{1}{2q_rb}+...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5194-when-is-a-rational-fraction-a-convergent-of-an-infinite-continued-fraction.html

  • Graph, Drawing and Lined Paper

    circle.pdf circle2.pdf dot1cm.pdf dot1cmiso.pdf handwriting.pdf hexagon.pdf multisquare.pdf numbersq1000.pdf octagon.pdf triangle_000.pdf tridots.pdf

    https://astarmathsandphysics.com/tools-and-resources/graph-drawing-and-lined-paper.html

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