Trigonometry is the study of three sided figures or triangles. It analyzes the relationships between the lengths of their sides and the measure of it's angles. Trigonometry was established from a need to locate points and to measure distances on land and in space. http://images.encarta.msn.com/xrefmedia/a encmed/targets/illus/ilt/t048529a.gif http://www.keithjmacdonald.ca/mct4c_trig_ miniunit/mct4c_trig_master_html_ 30558729.jpg http://openlearn.open.ac.uk/file.php/2737/ldt_1 _006i.jpg 1
Applications of trigonometry What can you do with trig? Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics, trig is used primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trig is a very useful subject to know. Astronomy and geography Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy. This information was found at: http://www.clarku.edu/~djoyce/trig/apps.html 2
As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy (100 178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus' Ephemerides Astronomicae on his trips to the New World and used it to his advantage. Engineering and physics Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long. Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig. Mathematics and its applications Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications. Calculus, linear algebra, and statistics, in particular, use trigonometry and have many applications in the all the sciences. This information was found at: http://www.clarku.edu/~djoyce/trig/apps.html 3
The following information was found at: http://www.humboldt.edu/~dlj1/precalculus/images/unitcircle.html The Unit Circle is a tool used in understanding sines and cosines of angles found in right triangles. Its radius is exactly one unit in length, usually just called "one". The circle's center is at the origin, and its circumference comprises the set of all points that are exactly one unit from the origin while lying in the plane. Click the following link to see how triangles are used in the animation. Y http://www.keymath.com/x3359.xml The radius equals one unit in the unit circle, and the center of the circle is the origin. Can you name 4 points on the unit circle? Center (0,0) radius = 1 unit X ( 1,0) (1,0) (0, 1) (0,1) 4
To make an angle on the unit circle: There are two sides of an angle. Initial side of an angle is a fixed ray along the x axis. Terminal side of an angle is a ray that does the moving in either a counterclockwise direction if the angle is positive or clockwise if the angle is negative. An angle is in standard position if the vertex is the origin and the initial side is on the positive side of the x axis. Quadrantal Angle in standard position has it's terminal side on either the x or y axis. List 5 examples. 5
The Greek Letter Theta θ θ is a Greek letter that we use as a variable to represent angles in Trigonometry. 6
Goal: Make a unit circle to use as a trigonometry trainer for class. 1) Take your paper plate and fold it in half both ways. 2) Use a ruler to trace over the folds with a Pencil. 3) Make one of the lines the x axis and the other the y axis. 4) On the top edges of the plate, mark quadrants I IV. 5) Trace the round part of the plate on the outer ridge so that the circumference of the unit circle is clearly shown. 6) Label Unit Circle Trig Trainer at the top of your paper plate. 7) Label the four points where the unit circle intersects the x and y axis. 8) Label which way angles open on the unit circle. (Positive angles open counterclockwise and negative angles open clockwise) 9) Label the first 4 Quadrantal angles (positive and negative) 10) State if the x and y values are positive or negative in each quadrant. 11) Start the 45 degree family. 12) Continue with the 30 and 60 degree families. 7
45 Degree family of triangles 45 o Note: The radius of the unit circle = Make a 45 degree angle in quadrant I of your trig trainer and draw an altitude to the x axis. Find the length of the base and height of the 45 degree triangle. 45 o What type of triangle is this? 8
45 Degree family of triangles R Y axis (X,Y) R (X,Y) X axis Reflect the 45 degree triangle from quadrant I to quadrants II, III, and IV. Label the new points where the triangles meet the unit circle on your trig trainer. 45 o 9
45 Degree family of triangles Name some of the angles in the 45 degree family o 45 o 45 45 o 10
30 Degree family of triangles Make a 30 degree angle in quadrant I of your trig trainer and draw an altitude to the x axis. 30 o Find the length of the base and height of the 30 degree triangle on the next page. 30 o 11
X= radius of unit circle = 1 unit X = 12
30 Degree family of triangles R Y axis (X,Y) R (X,Y) X axis Reflect the 30 degree triangle from quadrant I to quadrants II, III, and IV. Label the new points where the triangles meet the unit circle on your trig trainer. o 30 o 30 30 o 30 o 13
30 o 30 o Unit Circle Trig (Paper plate lessons).notebook 30 Degree family of triangles Name some of the angles in the 30 degree family 30 o 14
60 Degree family of triangles Make a 60 degree angle in quadrant I of your trig trainer and draw an altitude to the x axis. 60 o Find the length of the base and height of the 60 degree triangle. Hint: You have already done the work for this triangle. 60 o 15
60 o Unit Circle Trig (Paper plate lessons).notebook 60 Degree family of triangles R Y axis (X,Y) R (X,Y) X axis Reflect the 60 degree triangle from quadrant I to quadrants II, III, and IV. Label the new points where the triangles meet the unit circle on your trig trainer. 60 o 60 o 60 o 16
60 o Unit Circle Trig (Paper plate lessons).notebook 60 Degree family of triangles Name some of the angles in the 60 degree family 60 o 60 o 17
Final observations on how to use your trig trainer: Cos (θ) = X value on the unit circle Sin (θ) = Y value on the unit circle Remember that θ is a Greek letter used to represent an angle. Ex: Evaluate the Cos (135) You could look at your trig trainer for your answer. Look at the terminal side of the 135 degree angle and notice where it meets the circle. Since Cos is the x value on the unit circle the answer would be Ex: Evaluate Sin (330) Ex: Evaluate Sin ( 60) 18
II Y axis I Pos Angles X axis Neg Angles III Cos(θ) = X Sin(θ) = Y IV 19
θ 0 30 45 60 90 Cos (θ) X Sin(θ) Y Tan (θ) Y/X Use your calculator to help you fill in the chart above. Use exact values. 20
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