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Math Basics for Robotics β Trigonometry Long Wang 1 Triangles Q1: What does a triangle have? 2 Triangles π π1 π Q1: What does a triangle have? π π 3 vertices: π, π, π 3 sides: π1 , π2 , π3 3 angles (interior): π, π, π π2 π3 π π 3 Triangles π π3 π Q1: What does a triangle have? π π 3 vertices: π, π, π 3 sides: π1 , π2 , π3 3 angles (interior): π, π, π π1 π2 π π ππ₯π‘πππππ 4 Interior Angle Addition π π1 π Fact #1 to remember: π π π2 π3 π The sum of the measures of the interior angles of a triangle is always 180 degrees π π + π + π = 180β 5 Centers of a Triangle - Centroid Def. #1 to remember: π π1 π What is a centroid? Think it as center of mass, geometric center, or βaverageβ position of all the vertices. π π The centroid of a triangle the intersection of the three medians. π2 π3 π π 6 Centers of a Triangle - Centroid Def. #1 to remember: π π1 π What is a centroid? Think it as center of mass, geometric center, or βaverageβ position of all the vertices. π π The centroid of a triangle the intersection of the three medians. π2 π3 π π 7 Example 1 π π1 π π π 1) Pick arbitrary 3 vertices; 2) Connect them with 3 edges; 3) Label vertices, edges and angles; 4) Measure all of them. 5) Find the centroid of your triangle π2 π3 π π 8 Special Triangle β Right Triangle π π3 π π1 π π π2 A right triangle is a triangle in which one angle is a right angle (that is, a 90-degree angle). π Hypotenuse - the side opposite to the right angle. Legs - the sides adjacent to the right angle. Adjacent/opposite leg β need to pick an angle first. 9 Special Triangle β Right Triangle π π3 π π1 π π π2 A right triangle is a triangle in which one angle is a right angle (that is, a 90-degree angle). π Hypotenuse - π3 Legs - π1 , π2 For angle π, the adjacent leg is π2 , and the opposite leg is π1 . 10 Special Triangle β Right Triangle π Fact #2 to remember: π3 π π1 π and π are complementary. π π π2 π Complementary angles are angle pairs whose measures sum to one right angle π + π = 90β 11 Special Triangle β Right Triangle π Fact #2 to remember: π3 π π1 π and π are complementary. π π π2 π Complementary angles are angle pairs whose measures sum to one right angle π + π = 90β Quiz: what was the previous fact to remember? Can we use it to prove this fact? 12 Special Triangle β Right Triangle π π3 π π π π2 Fact #3 to remember: π1 The square of the hypotenuse is equal to the sum of the π squares of the other two sides. βPythagorean Theoremβ π12 + π22 = π32 13 Example 2 π π3 π π1 π π π2 π 1) Use ruler/triangle/grid paper to draw a right triangle whose legs are: β’ 3 and 4 β’ 5 and 12 β’ 8 and 15 2) Measure the length of hypotenuse 3) Are your findings consistent with Fact #3? 14 Cosine, Sine, Tangent π π π β’ Sine = Opposite / Hypotenuse β’ Cosine = Adjacent / Hypotenuse β’ Tangent = Opposite / Adjacent β’ sin π = π β β’ cos π = π β β’ tan π = π/π Reminder: sine, cosine, tangent are defined only in right triangles 15 Example 3 π π3 π π1 π π Given β’ π1 = 1 β’ π2 = 2 Solve π2 π β’ cos π=? β’ π=? 16 Example 3 π π3 π π1 Given β’ π1 = 1 β’ π2 = 2 π π π2 π Solve β’ π3 = π12 + π22 = 5 β’ cos π = π1 π3 = 1/ 5 1 β’ π = arctan(π1 /π2 ) = arctan = 26.6β 2 17 Special Triangle - Equilateral Fact #4 to remember: Special properties of equilateral triangle β’ All three sides have equal lengths β’ Each angle = 60β β’ Median = Altitude = Bisector 18 Special Triangle - Equilateral Fact #4 to remember: Special properties of equilateral triangle β’ All three sides have equal lengths β’ Each angle = 60β β’ Median = Altitude = Bisector β’ =30β , =60β 19 Special Triangle - Equilateral Fact #4 to remember: Special properties of equilateral triangle β’ All three sides have equal lengths β’ Each angle = 60β β’ Median = Altitude = Bisector β’ =30β , =60β Example 4 In the equilateral triangle above, if the length of each side is 10 cm, what is the length of the red line segment. 20 Law of Cosine Law of Cosine π2 = π 2 + πΏ2 β 2ππΏ cos π How do we use this equation? 21 Law of Cosine Law of Cosine π2 = π 2 + πΏ2 β 2ππΏ cos π OR πΏ2 + π 2 βπ2 cos π = 2ππΏ Law of Cosine tells us: 1) the third side can be solved if we know two sides and the angle between them 2) Each angle can be solved if we know all three sides 22 Example 5 π=2 πΏ=3 Solve π =? π=4 23 Example 5 π=2 πΏ=3 Solve π =? π=4 Solve β’ β’ π 2 +π2 βπΏ2 22 +42 β32 cos π = = 2ππ 2β2β4 11 π = arccos = 46.6β 16 = 11 16 24 Coordinates system A coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point. 25 Coordinates system (-1,3) (3,1) A coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point. (3,-2) 26 Coordinates system Def. #2 to remember: (-1,3) (3,1) The two axes divide the plane into four infinite regions, called quadrants. (3,-2) 27 Coordinates system Def. #2 to remember: (-1,3) II (-,+) I (+,+) (3,1) III (-,-) IV (+,-) The two axes divide the plane into four infinite regions, called quadrants. 28 Coordinates system π¦ π¦1 π Q1: How to find the coordinates (numbers) of a point that is not on the grid? (π₯1 , π¦1 ) π₯1 π₯ 29 Coordinates system π¦ π¦1 π Q1: How to find the coordinates (numbers) of a point that is not on the grid? (π₯1 , π¦1 ) π₯1 Q2: How to find the distance from origin to point (π₯1 , π¦1 )of a point that is not on the grid? π₯ 30 Coordinates system π¦ π¦2 π¦1 π Q1: How to find the coordinates (numbers) of a point that is not on the grid? (π₯2 , π¦2 ) (π₯1 , π¦1 ) π₯1 π₯2 π₯ Q2: How to find the distance from origin to point (π₯1 , π¦2 )of a point that is not on the grid? Q3: How to find the distance from point (π₯1 , π¦2 ) to point (π₯2 , π¦2 )? 31 Coordinates system π¦ (π₯2 , π¦2 ) π¦2 |π¦2 β π¦1 | π¦1 (π₯1 , π¦1 ) |π₯2 β π₯1 | π π₯1 π₯2 π₯ Solutions β’ πΏ1 = π₯12 + π¦12 β’ πΏ12 = π₯2 β π₯1 2 + π¦2 β π¦1 2 32 Example 6 π¦ Given the side length π, find the coordinates of three vertices. The coordinate system origin is at the centroid, and the y axis is aligned with one median. π₯ 33 Example 6 π¦ (π₯2 , π¦2 ) β’ π₯2 = 0 β’ π¦2 = π π₯ (π₯3 , π¦3 ) π 2 β’ π₯3 = βπ cos 30β β’ π¦3 = βπ sin 30β π β’ π = cos 30β 2 π 2 (π₯1 , π¦1 ) β’ π₯1 = π cos 30β β’ π¦1 = βπ sin 30β π β’ π = cos 30β 2 34 Redefine cosine, sine, tangent β unit circle Question: How do we define cosine, sine and tangent, if the angle is bigger than 90β , or even bigger than 180β ? 35 Redefine cosine, sine, tangent β unit circle A unit circle is a circle with a radius of one. π¦ π₯ 36 Redefine cosine, sine, tangent β unit circle A unit circle is a circle with a radius of one. (π₯, π¦) π β’ π = 45β π₯ β’ cos π = β’ sin π = π π¦ π 37 Redefine cosine, sine, tangent β unit circle A unit circle is a circle with a radius of one. β’ π = 135β π₯ β’ cos π = (π₯, π¦) π β’ sin π = π π¦ π 38 Redefine cosine, sine, tangent β unit circle A unit circle is a circle with a radius of one. β’ π = 225β π₯ β’ cos π = (π₯, π¦) π β’ sin π = π π¦ π 39 Redefine cosine, sine, tangent β unit circle A unit circle is a circle with a radius of one. β’ π = 315β π₯ β’ cos π = (π₯, π¦) π β’ sin π = π π¦ π 40 Special lines of a Triangle π π1 π π π Median a line segment joining a vertex to the midpoint of the opposing side. π2 π3 π Altitude a line segment through a vertex and perpendicular to the opposing side. π 41
