Page 92 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
3. Solving Trigonometric Equations
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
Sample problem: The population size, P, of owls (predators) in a certain region can be modelled by the function
P(t) = 1000 + 100 sin (π t ), where t represents 12
the time in months. The population size, p, of mice (prey) in the same region is given by
p(t) = 20 000 + 4000 cos (π t ). Sketch the 12
graphs of these functions, and pose and solve problems involving the relationships between the two populations over time.
90
2.4 determine the amplitude, period, and phase shift of sinusoidal functions whose equations are given in the form f(x) = a sin (k(x – d)) + c or f(x) = acos(k(x – d)) + c, with angles expressed in radians
2.5 sketch graphs of y = a sin (k(x – d)) + c and
y = acos(k(x – d)) + c by applying trans- formations to the graphs of f (x) = sin x and
f (x) = cos x with angles expressed in radians, and state the period, amplitude, and phase shift of the transformed functions
Sample problem: Transform the graph of f(x) = cos x to sketch g(x) = 3 cos (2x) – 1, and state the period, amplitude, and phase shift of each function.
2.6 represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians
Sample problem: A sinusoidal function has an amplitude of 2 units, a period of π, and a maximum at (0, 3). Represent the function with an equation in two different ways.
2.7 pose problems based on applications involv- ing a trigonometric function with domain expressed in radians (e.g., seasonal changes in temperature, heights of tides, hours of day- light, displacements for oscillating springs), and solve these and other such problems by using a given graph or a graph generated with or without technology from a table of values or from its equation
By the end of this course, students will:
3.1 recognize equivalent trigonometric expressions [e.g., by using the angles in a right triangle
torecognizethatsinxandcos(π2 –x)are
equivalent; by using transformations to
r e c o g n i z e t h a t c o s ( x + π2 ) a n d – s i n x a r e
equivalent], and verify equivalence using graphing technology
3.2 explore the algebraic development of the compound angle formulas (e.g., verify the formulas in numerical examples, using tech- nology; follow a demonstration of the alge- braic development [student reproduction of the development of the general case is not required]), and use the formulas to determine exact values of trigonometric ratios [e.g.,
determining the exact value of sin (π )by 12
first rewriting it in terms of special angles as sin (π4 – π6 )]
3.3 recognize that trigonometric identities are equations that are true for every value in the domain (i.e., a counter-example can be used to show that an equation is not an identity), prove trigonometric identities through the application of reasoning skills, using a variety
of relationships (e.g., tan x = sin x ; cos x
sin 2 x + cos 2 x = 1; the reciprocal identities; the compound angle formulas), and verify identities using technology
Sample problem: Use the compound angle formulas to prove the double angle formulas.
3.4 solve linear and quadratic trigonometric equa- tions, with and without graphing technology, for the domain of real values from 0 to 2π, and solve related problems
Sample problem: Solve the following trigono- metric equations for 0 ≤ x ≤ 2π, and verify by graphing with technology: 2 sin x + 1 = 0;
2 sin2 x + sin x – 1 = 0; sin x = cos 2x;
c o s 2 x = 12 .