• Vectors and Scalars

    adding vectors. Adding a force of 3N directed North to a force of 5N directed north east is best illustrated by drawing a triangle. We construct a parallelogram as shown. Now draw in the diagonal so that it represents the sum of the two vectors. We can...

    https://astarmathsandphysics.com/o-level-physics-notes/295-vectors-and-scalars.html

  • Constructing an Angle of Thirty Degrees

    draw another arc that crosses the first arc as shown. The point where the arcs cross is the third vertex of an equilateral triangle, after A and B. Join up the vertices. Each angle in the triangle is 60 Degrees. Label the top vertex C. Now bisect the...

    https://astarmathsandphysics.com/igcse-maths-notes/458-constructing-an-angle-of-thirty-degrees.html

  • Proof of Formula for Chord Bisecting Diameter at Right Angles

    For a chord bisecting a diameter of a circle at right angles as shown below, To prove it complete the triangle with one side as diameter to give the diagram below. The triangles and are similar, since both are right angled and angle If angle then from...

    https://astarmathsandphysics.com/igcse-maths-notes/498-proof-of-formula-for-chord-bisecting-diameter-at-right-angles.html

  • Proof of Formula Between Intersecting Chords

    For the chords drawn in the circle below, To prove it complete the triangles and draw an extra chord from A to D and an extra chord from B to C as shown below. From the Sine Rule for triangle ABO, and for triangle ODB, so that and C and D are equal...

    https://astarmathsandphysics.com/igcse-maths-notes/501-proof-of-formula-between-intersecting-chords.html

  • Stretch With Invariant x Axis

    the x or y direction. Stretches are defined in terms of a stretch factor and an invariant line. In the diagram above, triangle ABC is stretched to form the triangle A'B'C'. The invariant line is shown with arrows. Every point on the line is unmoved. If...

    https://astarmathsandphysics.com/igcse-maths-notes/528-stretch-with-invariant-x-axis.html

  • Stretch With Invariant y Axis

    the x or y direction. Stretches are defined in terms of a stretch factor and an invariant line. In the diagram above, triangle ABC is stretched to form the triangle A'B'C'. The invariant line is shown with arrows. Every point on the line is unmoved. If...

    https://astarmathsandphysics.com/igcse-maths-notes/529-stretch-with-invariant-y-axis.html

  • Constructing an Angle of Thirty Degrees

    draw another arc that crosses the first arc as shown. The point where the arcs cross is the third vertex of an equilateral triangle, after A and B. Join up the vertices. Each angle in the triangle is 60 Degrees. Label the top vertex C. Now bisect the...

    https://astarmathsandphysics.com/gcse-maths-notes/572-constructing-an-angle-of-thirty-degrees.html

  • Number Sequences and Shapes

    Consider the number sequence 1, 3, 6, 10... The nth term of this number sequence is We can picture the number sequences as triangles made up of dots. The nth triangle number is the number of dots in the nth triangle. Similarly, consider the number...

    https://astarmathsandphysics.com/gcse-maths-notes/614-number-sequences-and-shapes.html

  • Properties of the Modulus

    and for all for all for all and for all for all and for all and for all and all inegers for all and This is known as the triangle ineqaulity. for all and This is known as backwards form of the triangle ineqaulity. is the diatance on the real number line...

    https://astarmathsandphysics.com/ib-maths-notes/functions/978-properties-of-the-modulus.html

  • Proof That a Convex n – Sided Polygon Has n(n-3)/2 Diagonals

    A diagonal is a straght line in the interior of a polygon that goes from one vertex to another. A triangle has no diagonals, while a square has two and a pentagon has six. Each vertex is connected by edges of the polygon to two other vertices, so...

    https://astarmathsandphysics.com/ib-maths-notes/proofs/1082-proof-that-a-convex-n-sided-polygon-has-n-n-3-2-diagonals.html

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

    The compound angle formula can be proved using simple trigonometry. Consider the triangles below. From this diagram we can deduce the three extra angles in the triangle below. Then the sides and finally Using the whole triangle gives Multiplying both...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1133-proof-of-compound-angle-formula-sin-a%20b.html

  • The Ambiguous Case

    The ambiguous case arises when using the Sine Rule to find an angle in a triangle. It occurs because the Sin function is symmetric about 90°, so that When we solve for there is an acute solution, and an obtuse solution, Example: Find the angle A in the...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1142-the-ambiguous-case.html

  • Finding the Bearing a Plane Must Fly When Blown Off Course By Wind

    pilot must steer the plane into the wind slightly to follow the correct path over the ground. We can find an angle in the triangle by drawing in am extra line, as shown below. ABCD is a Z shape with Ab parallel to DC so angle ABC= angme BCD then the...

    https://astarmathsandphysics.com/ib-maths-notes/vectors-lines-and-planes/1156-finding-the-bearing-a-plane-must-fly-when-blown-off-course-by-wind.html

  • The Distance Between Two Points

    A and B in the diagram below. We draw a straight line between A and B, and make this rhe hypotenuse of a right angled triangle by drawing a horizontal line from A to a point directly below B and another line down from B to a point right of A. This...

    https://astarmathsandphysics.com/ib-maths-notes/vectors-lines-and-planes/1177-the-distance-between-two-points.html

  • Finding the Intercept, Gradient and Area Under the Graph

    in the x – value. For a straight line graph the gradient is constant. For example, the gradient below is Notice the triangle drawn is as large as possible. The graph illustrates Hooke's Law. It must be noted that the gradient has units derived from the...

    https://astarmathsandphysics.com/ib-physics-notes/measurements-units-uncertainty-and-principles/1341-finding-the-intercept-gradient-and-area-under-the-graph.html

  • Bodies on Slopes Toppling

    on a slope which makes an angle %theta with the horizontal. If a horizontal force is applied to the left at the top of the triangle, assuming the plane is rough enough so that the prism will topple before it slips, it will begin to topple when the...

    https://astarmathsandphysics.com/a-level-maths-notes/m3/3630-bodies-on-slopes-toppling.html

  • Pyramidal Numbers

    numbers are constructed starting with a single dot, then a triangle with a dot at each vertex, then a larger triangle with a dot between each two dots and so one. If these triangles are assembled into a pyramid, then the numbers of dots are called...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5123-pyramidal-numbers.html

  • Maximum Area of Rectangle Inscribed in Semicircle

    is the maximum area of a rectangle that can be inscribed in a semicircle of radius \[r\] ? Let the triangle OBC subtend an angle \[\theta\] at the centre of the circle. The triangles ABO abd CDO have the same area by symmetry, and together subtend an...

    https://astarmathsandphysics.com/a-level-maths-notes/c2/5322-maximum-area-of-rectangle-inscribed-in-semicircle.html

  • Volume of Solid Formed by Circle in xy P;ane and Lines To A Raised Diameter

    the circle at \[(x, - \sqrt{25-x^2} )\] to the point on the circle at \[(x, \sqrt{25-x^2} )\] , and an vertical isosceles triangle drawn between these two points and a point on the raised diameter. In this way a solid is formed. What is the diameter of...

    https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5362-volume-of-solid-formed-by-circle-in-xy-p-ane-and-lines-to-a-raised-diameter.html

  • Proof That Tangents Drawn From a Point to a Circle are Equal in Length

    The theorem is illustrated below. Proof: Construct the triangles and by drawing radii as below. since both are radii of the circle and is common to both. Further, angle since these are between a tangent and a radius. From Pythagoras theorem, so

    https://astarmathsandphysics.com/gcse-maths-notes/634-proof-that-tangents-drawn-from-a-point-to-a-circle-are-equal-in-length.html

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