Trigonometry

Essex Street Academy

Consortium school

Trigonometry is a one semester class during which students investigate the relationships between the sides and angles of triangles. They use trigonometric functions and their inverses to describe those relationships, the relationships of angles and coordinates on the unit circle, and the motion of waves. Students explore when, why, and how trigonometry can be used to solve practical problems and model real world phenomena.

This course emphasizes the following skills: applying trigonometry to solve for missing parts of a triangle, deriving trig formulas and identities, mapping the unit circle, and graphing trig functions as periodic waves to understand how manipulating their equations affects the amplitude, frequency, and vertical shift of the graph.

This course highlights the relationship between the different applications of trigonometry: how unit circle trigonometry is defined by triangle trigonometry and how the sine curve is defined by trigonometry of the unit circle.

Students learn these concepts and skills, and demonstrate their understanding, through projects and mini-projects which require hands on exploration.

Media Used: 
Graphing calculators
Significant Assignments: 

(1) By drawing and measuring, students investigate ratios of triangle sides with given angles in order to understand why the sine, cosine, or tangent of a particular angle is a fixed ratio. They then solve and create problems which require finding missing parts of a triangle given minimal information.

(2) Students use a clinometer and triangle trigonometry to find the height of a building or lamp post outside the school. They compare results from this with their prior estimates and with other mathematical methods of finding unknown lengths. Their write up includes a scale model and a description of their procedure.

(3) Each student presents a different proof of the Pythagorean theorem to the class. They then use the Pythagorean theorem to find ratios of sides in "special angle triangles" in simplest radical form.

(4) Students find the coordinates, in exact form, of points along the unit circle by drawing right triangles with the radius as the hypotenuse. They also use the circle and its subdivisions to find the radian measure of each special angle and formulate a rule for converting between radians and degrees.

(5) Students sketch a graph of y=sin(a) based on their knowledge of triangle and circle trigonometry by measuring the height of various points at intervals along the unit circle. The height, or y-coordinate, is sin(a) where a is the measure of the central angle. They then translate the height to a rectangular coordinate grid in order to discover the periodic nature of the sine curve. Students use the graph to find the sine of quadrantal angles (the x-intercepts, the maximums and minimums); determine and define the period, amplitude, and range; determine in which quadrants the value is increasing or decreasing; use the graphing calculator to verify their graphs. After students experiment with graphing calculators changing a,b, and d in the equation y=a*sin(bx)+d, they will explain how each one changes the appearance of the graphs in terms of amplitude, frequency, period, and vertical shift.

Significant Activities or Projects: 

(1) Ferris Wheel Project: Student convert situational descriptions involving periodic motion (of riders on a Ferris wheel) into equations and graphs. They create sine (or cosine) equations and graphs to model the height of riders and discuss the contextual meaning of frequency, period, amplitude, and vertical shift as they appear in the equation y=a*sin(bx)+d. They discuss how these components must change to describe changes in a Ferris wheel's speed, size, height, rotational direction, or mounting place.

(2) Students model and graph tidal patterns using trigonometric functions (using a reasonable scale and "window"). Using estimation based on the graph, and then increasingly more precise algebraic methods, they calculate the time and position of high tide, low tide, and average tide, as well as how frequently the tide comes in. Finally, they calculate how much time a sand-castle maker would have to build and admire a castle (before the tide submerges it) at various points on the beach. Students extend the problem in ways of interest to them.

(3) Students collect data on sunrise and sunset times for various locations around the world. They model data from a particular place with a trigonometric function (using sine regression). They describe the characteristics of daylight hours in their location, including the dates of solstices and equinoxes, using the characteristics of the trigonometric function they develop. Each student compares data with two other students in the class and uses latitude and Earth's tilt to explain differences in amplitude and phase. Students have to defend their use of a sine regression and describe what each constant in their equation means in terms of daylight hours.They then assess how well their regression fits their data by comparing their calculations and graphs based on the regression with calculations and graphs based on the actual data. They use their work to predict daylight hours in the future and solve inequalities to make recommendations about if and when solar energy is viable in that location.

Sample PBATs: 
Students collect data on sunrise and sunset times for various locations around the world. They model data from a particular place with a trigonometric function (using sine regression). They describe the characteristics of daylight hours in their location, including the dates of solstices and equinoxes, using the characteristics of the trigonometric function they develop. Each student compares data with two other students in the class and uses latitude and Earth's tilt to explain differences in amplitude and phase. Students have to defend their use of a sine regression and describe what each constant in their equation means in terms of daylight hours.They then assess how well their regression fits their data by comparing their calculations and graphs based on the regression with calculations and graphs based on the actual data. They use their work to predict daylight hours in the future and solve inequalities to make recommendations about if and when solar energy is viable in that location.