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76 | eUreka Math algebra II StUdy gUIde
Lesson 5: Extending the Domain of Sine and Cosine to All Real Numbers
● (^) Students define sine and cosine as functions for all real numbers measured in degrees.
● (^) Students evaluate the sine and cosine functions at multiples of 30 and 45.
Lesson 6: Why Call It Tangent?
● (^) Students define the tangent function and understand the historical reason for its
name.
● (^) Students use special triangles to determine geometrically the values of the tangent
function for 30, 45, and 60 degrees.
Lesson 7: Secant and the Co-Functions
● (^) Students define the secant function and the co-functions in terms of points on the unit
circle. They relate the names for these functions to the geometric relationships among
lines, angles, and right triangles in a unit circle diagram.
● (^) Students use reciprocal relationships to relate the trigonometric functions to each
other and use these relationships to evaluate trigonometric functions at multiples of
30, 45, and 60 degrees.
Lesson 8: Graphing the Sine and Cosine Functions
● (^) Students graph the sine and cosine functions and analyze the shape of these curves.
● (^) For the sine and cosine functions, students sketch graphs showing key features, which
include intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maxima and minima; symmetries; end behavior; and periodicity.
Lesson 9: Awkward! Who Chose the Number 360, Anyway?
● (^) Students explore horizontal scalings of the graph of yx=°sin( ).
● (^) Students convert between degrees and radians.
Lesson 10: Basic Trigonometric Identities from Graphs
● (^) Students observe identities from graphs of sine and cosine basic trigonometric
identities and relate those identities to periodicity, even and odd properties, intercepts,
end behavior, and cosine as a horizontal translation of sine.
Topic B: Understanding Trigonometric Functions and Putting Them to Use
In Topic A, students developed the ideas behind the six basic trigonometric functions,
focusing primarily on the sine function. In Topic B, students use trigonometric functions to
model periodic behavior. We end the module with the study of trigonometric identities and
how to prove them.
Lesson 11 continues the idea started in Lesson 9 in which students graphed yk=°sin( x)
for different values of k. In Lesson 11, teams of students work to understand the effect of
changing the parameters A, ω, h, and k in the graph of the function yA=-(sin((wxh)))+k,
so that in Lesson 12 students can fit sinusoidal functions to given scenarios, which aligns with
F-IF.C.7e and F-TF.B.5. While Lesson 12 requires that students find a formula that precisely
models periodic motion in a given scenario, Lesson 13 is distinguished by nonexact modeling,
as in S-ID.B.6a. In Lesson 13, students analyze given real-world data and fit the data with an
appropriate sinusoidal function, providing authentic practice with MP.3 and MP.4 as they
debate about appropriate choices of functions and parameters.