Page 55 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

D. TRIGONOMETRIC FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. determine the values of the trigonometric ratios for angles less than 360o; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law;
2. demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions;
3. identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 determine the exact values of the sine, cosine, and tangent of the special angles: 0o, 30o, 45o, 60o, and 90o
1.2 determine the values of the sine, cosine, and tangent of angles from 0o to 360o, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)
1.3 determine the measures of two angles from 0o to 360o for which the value of a given trigonometric ratio is the same
1.4 define the secant, cosecant, and cotangent ratios for angles in a right triangle in terms of the sides of the triangle (e.g.,
sec A = ), and relate these ratios
to the cosine, sine, and tangent ratios (e.g.,
secA= 1 ) cos A
1.5 prove simple trigonometric identities, using the Pythagorean identity sin2x + cos2x = 1;
the quotient identity tanx = sinx ; and cos x
the reciprocal identities secx =
1 , cos x 1
1
cscx =
, and cotx =
sin x
tan x
Sample problem: Prove that 1 – cos2x = sinxcosxtanx.
1.6 pose problems involving right
triangles and oblique triangles in two- dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)
1.7 pose problems involving right triangles and oblique triangles in three-dimensional set- tings, and solve these and other such pro- blems using the primary trigonometric ratios, the cosine law, and the sine law
Sample problem: Explain how a surveyor could find the height of a vertical cliff that is on the other side of a raging river, using a measuring tape, a theodolite, and some trigonometry. Determine what the surveyor might measure, and use hypothetical values for these data to calculate the height of the cliff.
By the end of this course, students will:
2.1 describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay
of Fundy), given a numeric or graphical representation
TRIGONOMETRIC FUNCTIONS
1. Determining and Applying Trigonometric Ratios
hypotenuse
adjacent
2. Connecting Graphs and Equations of Sinusoidal Functions
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