• When to Use the Cosine Rule and When to Use the Sine Rule

    The Cosine Rule and the Sine Rule are both used to find sides and angles in triangles that are not right angled. For the triangle below, the Cosine Rule states The Sine Rule states Which rule to use depends on the combination of angles and sides you...

    https://astarmathsandphysics.com/gcse-maths-notes/690-when-to-use-the-cosine-rule-and-when-to-use-the-sine-rule.html

  • Young's Double Slit Experiment

    fringe the path difference between light from the two slits must be equal to one wavelength, Because is an angle in both triangles we may say approximately In general is the distance between successive fringes, the fringe spacing.

    https://astarmathsandphysics.com/o-level-physics-notes/300-young-s-double-slit-experiment.html

  • Nets and Surface Area

    flat surface and finding the areas of individual parts of the net, which are often simple shapes like rectangles, squares, triangles and circles. To find the area of the cuboid below, draw the net, with the length of each side, which looks like The area...

    https://astarmathsandphysics.com/igcse-maths-notes/488-nets-and-surface-area.html

  • Nets and Surface Areas

    flat surface and finding the areas of individual parts of the net, which are often simple shapes like rectangles, squares, triangles and circles. To find the area of the cuboid below, draw the net, with the length of each side, which looks like The area...

    https://astarmathsandphysics.com/gcse-maths-notes/613-nets-and-surface-areas.html

  • Proof of Compound Angle Formula sin(A-B)

    The compound angle formula can be proved using simple trigonometry. Consider the right angled triangle below with internal right angled triangles constructed with internal angles and With sides labelled and as shown below, we can find sides in terms of...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1132-proof-of-compound-angle-formula-sin-a-b.html

  • Definition of Terms Used For Lenses

    of the object, also equal to the ratio of the image and object distances: This is easily seen from the diagram below. The triangles OAB and BCI are similar, so and which rearranges to (1) .

    https://astarmathsandphysics.com/ib-physics-notes/optics-and-light/1391-definition-of-terms-used-for-lenses.html

  • Huygen's Principle

    ray to go from B to B' the reflected ray at A goes from A to A'. The speed is constant so BB'=AA'. AB' is common to triangles ABB' and AA'B' so∡BAB' equals ∡A'B'A, but ∡BAB' + i = A'B'A +r so i=r. Refraction In the time that the incident wavefront...

    https://astarmathsandphysics.com/ib-physics-notes/waves-and-oscillations/1489-huygen-s-principle.html

  • Proof That the Angle Between a Chord and a Tangent at the Point of Contact is Equal to an Angle in the Alternate Segment

    The theorem is illustrated below. Proof: Draw a diameter at as below. The angle and angle Using the triangle angle then by this theorem, Similarly angle

    https://astarmathsandphysics.com/gcse-maths-notes/635-proof-that-the-angle-between-a-chord-and-a-tangent-at-the-point-of-contact-is-equal-to-an-angle-in-the-alternate-segment.html

  • Ratios and Similar Shapes

    one shape onto the other. For example, consider the shape below: We may separate the shape into a smaller and a larger triangle, shown below. The enlargement from the smaller to the larger is (by considering the sides 6 and 11), hence

    https://astarmathsandphysics.com/gcse-maths-notes/642-ratios-and-similar-shapes.html

  • Vectors 2

    then we can find using this ratio. Suppose splits in the ratio 2:3, then is in the ratio 2/3 so that and We can label the triangle as below. Then For a slightly more complicated example, consider the parallelogram below, with and splits in the ratio 1:2...

    https://astarmathsandphysics.com/gcse-maths-notes/686-vectors-2.html

  • Volume

    The volume of a prism = the area of the cross-section × the length. So, for example, the volume of a cylinder area of a triangle = half * base * height so the volume of a triangular prism is half*base*height*length area of a circle ( is the radius of...

    https://astarmathsandphysics.com/gcse-maths-notes/687-volume.html

  • Working With Column Vectors

    vector notation since more information is included, and is preferable when working in the x – y plane. Suppose we have the triangle below. P splits AC in the ratio 1:2. The vector from A to B is and the vector from B to C is Then and

    https://astarmathsandphysics.com/gcse-maths-notes/691-working-with-column-vectors.html

  • The Ferrymans Problem

    pointed into the stream. The boat still travels at a speed through the water but is now the hypotenuse of the right angled triangle. The boat must point its bow to travel at an angle to the intended path. The effective speed of the boat is reduced from...

    https://astarmathsandphysics.com/o-level-physics-notes/242-the-ferrymans-problem.html

  • Sketching Inequalities and Finding the Region Satisfied By Inequalities

    above the line. All the lines are solid because points on the line may satisfy the inequalities. The region we want is the triangle in the middle, labelled R.

    https://astarmathsandphysics.com/o-level-maths-notes/347-sketching-inequalities-and-finding-the-region-satisfied-by-inequalities.html

  • Formulae

    from n possibilities in possible ways. You can line up r from n possibilities in different ways Binomial Theorem : For any triangle ∆ABC , The straight line passing through has gradient and equation The distance between two points, is A straight line...

    https://astarmathsandphysics.com/igcse-maths-notes/480-formulae.html

  • Ratios and Similar Shapes

    one shape onto the other. For example, consider the shape below: We may separate the shape into a smaller and a larger triangle, shown below. The enlargement from the smaller to the larger is (by considering the sides 6 and 11), hence

    https://astarmathsandphysics.com/igcse-maths-notes/508-ratios-and-similar-shapes.html

  • Symbols and Notation

    B –the complement of the set A – the set of all elements not in set A. –the Angle at A -the angle between CA and AB -the triangle whose vertices are A, B and C - the vector –the length or magnitude of the vector –the vector from point A to point B –the...

    https://astarmathsandphysics.com/igcse-maths-notes/532-symbols-and-notation.html

  • Working With Column Vectors

    vector notation since more information is included, and is preferable when working in the x – y plane. Suppose we have the triangle below. P splits AC in the ratio 1:2. The vector from A to B is and the vector from B to C is Then and

    https://astarmathsandphysics.com/igcse-maths-notes/548-working-with-column-vectors.html

  • Angles in a Regular Polygon

    are shown below. All the sides in a regular polygon are the same length and the interior angles are all the same. For a triangle they are all 60 degrees, for a square 90 degrees, for a pentagon 108 degrees... The external angles – the angle between a...

    https://astarmathsandphysics.com/gcse-maths-notes/556-angles-in-a-regular-polygon.html

  • Angles of Elevation

    below. The angle of elevation of the top of the tree from the groundis We can use the diagram to draw the right angled triangle below. The angle of elevation is the solution to Conversely we might want to find the angle of depression of the point A on...

    https://astarmathsandphysics.com/gcse-maths-notes/558-angles-of-elevation.html

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