MATHOMAT SENIOR TEMPLATE. A technical and creative drawing tool for senior secondary school students.

Transcription

1 SENIOR TEMPLATE A technical and creative drawing tool for senior secondary school students. Mathomat Senior has been specifically developed for Sketching and presentation of work by students from years 10 to 1. Features such as a unit circle and tangent line for trigonometry study, graphing curves and 3D soild sketch outline make Mathomat Senior an invaluable aid for projects and classwork, particularly for topis such as Trigonometry, Geometry & Measurement and Fractions & Graphs. 1

2 OLM - M FEATURES OF TANGENT SENIOR Sine and cosine curves On a unit circle, Accurate of various amplitudes Tangent of an angle = position of the extended angle arm on the tangent line. rulers. and periods. A tangent line scale is marked on the right edge of Mathomat Senior, and must be re-drawn as shown before the tangent of an angle can be read. Cylinder outline tan 50 = 1. (approx.) Prism cluster. Isometric drawing guides. Three different ellipses. Special triangles (1, 1, and 1,, 3). Pyramid outline. 10 graph curves Unit circle and tangent line. 5 circle sizes. 4 squares. Trig graph scale. The diagram above shows that tangent of 50 = 1. (approx.), or tan 30 = 1. (approx.). Quadratic graphs. A complete list follows on the opposite page.

3 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 4 TRIGONOMETRIC SENIOR SPECIFICATIONS SYMMETRY Symmetry properties of the trigonometric function my be determined easily using Mathomat Senior s unit circle, which Name has angles in terms of marked. Specifications 1 cosine/sine curve Period 4cm. Amplitude 1 cm The diagram below shows that sine/cosine curve Period 8cm. Amplitude 1 cm 3 sine/cosine curve Period 4cm. Amplitude cm 4 Circle.5 cm diameter / T P 5 Circle cm diameter 6 Cylinder Straight sides cm, width cm 7 Cone cm base,.5 cm height x y = a (exponential graph) Normal curve Circle 1 cm diameter 11 Rectangle 3 cm x cm 1 Parallelogram 3 cm, cm, Ohombus cm, Ellipse 4 cm x cm 15 Equilateral triangle Inscribes within circle 9 16 Square 3 cm x 3 cm 17 Triangle (1, 1, ) 3 cm, 3 cm, 45, 45, Sqaure pyramid Ellipse cm x 1 cm These 0 statements Parallelogram may be generalised to find expressions for 3 cm, cm, 60 1 y = tan x (tangent graph) 3 y = x (cubic graph) Similar 3 expressions Sqaure may be derived for the tangent function cm x 1 cm (These 4 tasks Sqaure are left to the reader) 1.5 cm x 1.5 cm 5 Sqaure 1 cm x 1 cm 6 SPECIAL TRIANGLES y = ½ x (quadratic graph) Shapes 17 and 3 are the special triangles commonly used to determine trig values of key angles as shown below. 7 y = x (quadratic graph) 8 9 y = x (quadratic graph) Ellipse 4 cm x 1 cm 30 Circle 3 cm diameter 31 3 Circle Triangle (1,, 3 ) 4 cm diameter 3 / H 3

4 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 9 OLM - M ANGLE THREE MEASUREMENT DIMENSIONAL AND GEOMETRY CONSTRUCTION Three dimensional solids maybe be sketched quickly and accurately using the shapes listed below. Use shapes 6, 19 Use shapes 7, 19 Use shapes 18 Even previously tricky reflex angles may be drawn and measured easily using Mathomat Senior s circular protractor. Use shapes 11, 1, 13 Use shapes 1, 19 Use shapes 3 PERPENDICULAR LINES Use shapes 5, 19 Use shapes 9, 31 Use the lines that divide Mathomat Senior vertically and horizontally as guides for constructing perpendicular lines, for example when Use drawing shapes 14, large 31 rectangles. PARALLEL LINES Use shapes 15, 9 Use the left edge ruler and straight edges of shapes 6 and 17 as shown to construct parallel lines 1 cm apart. 4

5 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 610 LARGE ISOMETRIC SCALE POLYGONS LINES Large These polygons allow may diagram be drawn like using the one the below letter to guides be drawn around with the relative outside ease. of the circular protractor. Simply mark each corner using the appropriate letter, and connect them. The following letters are used around the circular protractor. T - triangle, P - pentagon, H - hexagon, and O - octagon. For a square, use the 0, 90, 180 and 70 holes on the edge of the circular protractor. UNIT CIRCLE TRIGONOMETRY Definitions od sine, cosine and tangent are easily explained and visualised using the circular protractor which doubles as a unit circle. Sine and Cosine On a unit circle, Sine of an angle = y co-ordinate Cosine of an angle = x co-ordinate For example, the following diagram shows that Sine of 30 = 0.5 and Cosine of 30 = 0.87 (approx.) or, using mathematical shorthand, sin 30 = 0.5 and cos 30 = 0.87 (approx.) Isometric Lines 5

6 GRAPHS OF TRIGONOMETRIC FUNCTIONS Mathomat Senior contains three sine/cosine curves. Each one can be used to sketch any sine or cosine graph by labeling the amplitude and period appropriately. For example, the following four graphs show four different functions, but were all drawn using shape1. Of course shapes and 3 may also be used to highlight different periods or amplitudes. Again, the choice of axis can force; a curve to match a particular equation. 6

7 TANGENT GRAPHS Shape 1 may maybe used to sketch tangent graphs. Again, an appropriate scale needs to be chosen to ensure the curve matches the specified equation. GRAPHS OF OTHER FUNCTIONS Mathomat Senior contains graphing curves for most of the major functions met in senior Maths classes. Curves and the shapes used to sketch then appear below. 7

8 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 9 OLM - M THREE DIMENSIONAL GEOMETRY Three dimensional solids maybe be sketched quickly and accurately using the shapes listed below. Use shapes 6, 19 Use shapes 7, 19 Use shapes 18 Use shapes 11, 1, 13 Use shapes 1, 19 Use shapes 3 Use shapes 5, 19 Use shapes 9, 31 Use shapes 14, 31 Use shapes 15, 9 8

9 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 610 LARGE ISOMETRIC SCALE POLYGONS LINES Large These polygons allow may diagram be drawn like using the one the below letter to guides be drawn around with the relative outside ease. of the circular protractor. Simply mark each corner using the appropriate letter, and connect them. The following letters are used around the circular protractor. T - triangle, P - pentagon, H - hexagon, and O - octagon. For a square, use the 0, 90, 180 and 70 holes on the edge of the circular protractor. UNIT CIRCLE TRIGONOMETRY Definitions od sine, cosine and tangent are easily explained and visualised using the circular protractor which doubles as a unit circle. Sine and Cosine On a unit circle, Sine of an angle = y co-ordinate Cosine of an angle = x co-ordinate For example, the following diagram shows that Sine of 30 = 0.5 and Cosine of 30 = 0.87 (approx.) or, using mathematical shorthand, sin 30 = 0.5 and cos 30 = 0.87 (approx.) Isometric Lines 9

10 OLM - M TANGENT On a unit circle, Tangent of an angle = position of the extended angle arm on the tangent line. A tangent line scale is marked on the right edge of Mathomat Senior, and must be re-drawn as shown before the tangent of an angle can be read tan 50 = 1. (approx.) The diagram above shows that tangent of 50 = 1. (approx.), or tan 30 = 1. (approx.). 10

11 OLM - Mathomat Senior Book redo _Layout 1 5/10/11 8:1 PM Page 4 TRIGONOMETRIC SENIOR SPECIFICATIONS SYMMETRY Symmetry properties of the trigonometric function my be determined easily using Mathomat Senior s unit circle, which Name has angles in terms of marked. Specifications 1 cosine/sine curve Period 4cm. Amplitude 1 cm The diagram below shows that sine/cosine curve Period 8cm. Amplitude 1 cm 3 sine/cosine curve Period 4cm. Amplitude cm 4 Circle.5 cm diameter / T P 5 Circle cm diameter 6 Cylinder Straight sides cm, width cm 7 Cone cm base,.5 cm height x y = a (exponential graph) Normal curve Circle 1 cm diameter 11 Rectangle 3 cm x cm 1 Parallelogram 3 cm, cm, Ohombus cm, Ellipse 4 cm x cm 15 Equilateral triangle Inscribes within circle 9 16 Square 3 cm x 3 cm 17 Triangle (1, 1, ) 3 cm, 3 cm, 45, 45, Sqaure pyramid Ellipse cm x 1 cm These 0 statements Parallelogram may be generalised to find expressions for 3 cm, cm, 60 1 y = tan x (tangent graph) 3 y = x (cubic graph) Similar 3 expressions Sqaure may be derived for the tangent function cm x 1 cm (These tasks 4 Sqaure are left to the reader) 1.5 cm x 1.5 cm 5 Sqaure 1 cm x 1 cm 6 SPECIAL TRIANGLES y = ½ x (quadratic graph) Shapes 17 and 3 are the special triangles commonly used to determine trig values of key angles as shown below. 7 y = x (quadratic graph) 8 9 y = x (quadratic graph) Ellipse 4 cm x 1 cm 30 Circle 3 cm diameter 31 3 Circle Triangle (1,, 3 ) 4 cm diameter 3 / H 11

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