• Park Links

    Here are links to some of my favorite parks. The weblinks component provides an easy way to make links to external sites that are consistently formatted and categorised. You can create weblinks from the front end of your site.

    https://astarmathsandphysics.com/component/weblinks/category/18-sample-data-weblinks/31-park-links.html?Itemid=829

  • Trigonometry in Three Dimensions

    Ordinary trigonomtry is possible wherever there are straight lines. The diagram below shows a rectangular block. I want ro find the distance AG and the angles AGB and BGD. To find the distance AG, use Pythagoras Theorem in three dimensions. AB, BC and...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1151-trigonometry-in-three-dimensions.html

  • Formulae Used in Trigonometry

    There must be any number of trigonometric identities that can be proved. Really the only way to study them is to do lots of examples. It helps first to have a summary of the equations. There are five very important equations which can also be used to...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1123-formulae-used-in-trigonometry.html

  • Simple Trigonometry - Finding Angles in Right Angled Triangles

    Simple trigonometry can be used with right angled triangles to find an angle given two sides, or a side given a side and one of the interior angles other than the right angle, With the sides of the triangle as labelled above, we can use one of the...

    https://astarmathsandphysics.com/o-level-maths-notes/341-simple-trigonometry-finding-angles-in-right-angled-triangles.html

  • Trigonometry - Right Angled Triangles - Finding a Side

    There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we have: If we have the angle shown, we label the sides for this angle. Opposite = is the side...

    https://astarmathsandphysics.com/igcse-maths-notes/540-trigonometry-right-angled-triangles-finding-a-side.html

  • Trigonometry - Right Angled Triangles - Finding a Side

    There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we have: If we have the angle shown, we label the sides for this angle. Opposite = is the side...

    https://astarmathsandphysics.com/gcse-maths-notes/682-trigonometry-right-angled-triangles-finding-a-side.html

  • Trigonometry and Right Angled Triangles - Finding an Angle

    There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we choose or have been given: Given we have chosen the angle we label the sides for this angle....

    https://astarmathsandphysics.com/gcse-maths-notes/681-trigonometry-and-right-angled-triangles-finding-an-angle.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

  • Formula for the Area of a Triangle

    The area of a triangle is where is the base and is the perpendicular height. This formula is a bit artififical because we can take any side as the base and the height will be the perendicular distance of the opposite vertex from this chosen base. There...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1124-formula-for-the-area-of-a-triangle.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

  • The Tetrahedron

    The tetrahedron is a four sided shape. Each side is a equilateral triangle. We can find angles in the tetrahedron using only the properties of right angled triangles – simple triogonometry and Pythagoras theorem. If the length of each side is 1, the...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1148-the-tetrahedron.html

  • Quadratic Trigonometric Equations

    A quadratic trigonometric equation is any equation with a or term, or a or term, since these become quadratic on using the appropriate double angle identity. The solutions of trigonometric equations can be found by factorisation or using the quadratic...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1137-quadratic-trigonometric-equations.html

  • Radians and Degrees

    A full turn is equivalent to a turn through 360° and half a turn is equivalent to a turn through 180°. A full turn is also equivalent to a turn through radians or and half a turn is equivalent to a turn through (Remember that 360° or is the angle at...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1138-radians-and-degrees.html

  • Relationships Bewtween Values of Trigonometric Functions

    There are some very useful relationships between the values of the elementary trigonometric functions that rely on the symmetry of thise functions. The function is symmetrical in the line and This means that also has rotational symmetry about the point...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1140-relationships-bewtween-values-of-trigonometric-functions.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

  • Problem Solving With Areas

    When finding the area of a composite shape or part of a composite shape, this is often symetry that can be used. The diagram below shows three overlapping circles, each of radius The centres of the circles are at A, B and C respectively and each circle...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1129-problem-solving-with-areas.html

  • Simplifying Expressions With Trigonometric Functions of Inverse Trigonometric Functions

    We can simplify expressions with sines and cosines of inverse trigonometric functions using substitutions. To find \[sin(2 sin^{-1}x)\] substitute \[\theta = sin^{-1}x\] then \[sin(2 sin^{-1}x)=sin(2 \theta)=2 sin \theta cos \theta =2x \sqrt{1-x^2}\] (...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/5015-simplifying-expressions-with-trigonometric-functions-of-inverse-trigonometric-functions.html

  • Using the Multiple Angle Formulae to Find Values of Triginometric Functions

    Any A Level student should know the values of any trigonometric function for any angle that is a multiple of 30 or 45 degrees. For convenience they are given below. 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 0 0 -1 1 0 -1 0 0 1 -1 0 1 -1...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1152-using-the-multiple-angle-formulae-to-find-values-of-triginometric-functions.html

  • Equations With Reciprocal Trigonometric Functions

    The reciprocal trigonometric functions are and Given an equation of this sort we find the value of or first, then find the reciprocal of this to find the value of or then solve this or these equation using the normal method for finding multiple angles...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1150-trigonometric-equations-with-reciprocal-functions.html

  • Cones

    The apex of a cone is always the same distance, equal to the sant height, from the edge of the base. This means that when we cut the cone along a straight line from the base to the apex and unfold it, the shape formed is a sector of a circle. The...

    https://astarmathsandphysics.com/ib-maths-notes/trigonometry/1120-cones.html

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