Understand the unit circle as the origin of the trigonometric functions Understand the definition of a radian and know how to convert between radians and degrees as measures of angles Understand the definition of a trigonometric function (sine, cosine and tangent) for any angle and their transformations Understand how to use trigonometric functions to solve trig equations and to model and understand periodic phenomena
Conceptual Framework
Form representation, model, generalization in the context of Orientation in Space and Time
Statements of Inquiry
Understanding movement of position on the circumference of a circle will enable us to model repeating change
Factual
Where do the trig ratios exist in a circle on the coordinate plane?
Conceptual
How do we manipulate the trig ratios in a circle to understand repeating change?
Debatable
Can mathematical functions accurately model cyclical real-world phenomena?
Description
The focus of this unit will be on Trigonometric functions, their representation and application to model cyclical change. Students will review the fundamental concept of the trigonometric ratios and investigate their existence and behaviour within the unit circle. Students will understand radians as an angular unit of measure and derive simple trigonometric identities from the circle. Once familiar with the unit circle, students will derive the base trigonometric functions geometrically. Students will investigate transformations of these functions using the form \(f(x) = A\sin[B(x-C)] + D\) and \(f(x) = A\cos[B(x-C)] + D\). Students will practice finding solutions to trigonometric equations using both algebraic and graphical methods. Students will also apply skills and concepts to model real life situations including Ferris Wheels, tides, and phases of the moon.
Learning Outcomes
Use and understand radian measure
Appreciate that the most efficient and accurate way to convert between degrees and radians is to think in fractions of 180
Convert between radians and degrees
Develop, memorize and apply the special angles \(0^\circ ,\,\,30^\circ ,\,\,45^\circ ,\,\,60^\circ ,\,\,90^\circ \) and using radians \(0,\,\,\frac{\pi }{6},\,\,\frac{\pi }{4},\,\,\frac{\pi }{3},\,\,\frac{\pi }{2}\) to solve problems requiring exact values
Appreciate that the x-coordinate on the circumference of the unit circle is the cosine of the angle formed by the terminal ray and the x-axis and that the y-coordinate on the circumference of the unit circle is the sine of that angle
Apply reference angles and the CAST rule for the trig functions to find the sine, cosine, tangent of any angle
Derive the trigonometric functions on the domain \(\left( {0,\,\,360^\circ } \right)\) or \(\left( {0,\,\,2\pi } \right)\) from the unit circle.
Understand and apply the transformations of trig functions (amplitude, period, vertical and horizontal shifts)
Sketch the graphs of transformed trig functions on a given domain
Solve trig equations using algebraic and graphical methods in domains on a given domain
Find the equation of a trig function from its graph
Model periodic phenomena using trig functions
Use the unit circle to derive the identities \(\tan x = \frac{\sin x}{\cos x}\) and \(\sin^2x+\cos^2x=1\)
Use simple Identities to simplify expressions, solve equations and introduce proofs