

Unit 2 Algebra II 

Unit Topic and
Length: 20
days Building on their previous work with
functions, and on their work with trigonometric ratios and circles in
Geometry, in Unit 2 students will now use the coordinate plane to extend
trigonometry to model periodic phenomena.
In
Geometry students will have used basic trigonometric ratios to solve problems
involving right triangles. This unit will be the first introduction to the
concept of a radian as an angle measure. Students will understand the radian
measure of an angle as the length of the arc on the unit circle subtended by
the angle. Students will understand the unit circle and its usefulness to
extend trigonometric functions to all real numbers. Additionally, students
will prove the Pythagorean identity sin^{2}(θ) + cos^{2}(θ)
= 1, and use it in their work with angles, measures, and location. Work in
this unit will prepare students for more extensive graphing, interpreting,
and modeling of trigonometric functions, along with other functions, in Unit
3. (Mathematics Appendix A, p.3640, with
adjustments) ·
Extend the domain of
trigonometric functions using the unit circle. ·
Model periodic
phenomena with trigonometric function. ·
Prove and apply
trigonometric identities. 

Standards: FTF.A.1
Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.
FTF.A.2
Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures
of angles traversed counterclockwise around the unit circle.
FTF.A.3(+) Use special triangles to determine geometrically the values of
sine, cosine, tangent for /3,
/4 and /6, and use the unit circle to express the
values of sine, cosine, and tangent for
x, +x, and
2 x in terms of
their values for x, where x is any real number.
FTF.B.5
Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.★
FTF.C.8
Prove the Pythagorean identity sin^{2}(θ) + cos^{2}(θ)
= 1 and use it to find sin(θ), cos(θ), or tan(θ) given
sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
FTF.C.9(+) Prove the addition and subtraction formulas for sine, cosine, and
tangent and use them to solve problems.
Focus Standards of Mathematical Practice: MP.1 Make
sense of problems and persevere in solving them. MP.2 Reason
abstractly and quantitatively. MP.3
Construct viable arguments and critique the reasoning of others. MP.4 Model
with mathematics. MP.5 Use
appropriate tools strategically. MP.6 Attend
to precision. MP.7 Look for
and make use of structure. MP.8 Look for
and express regularity in repeated reasoning. Instructional Outcomes: Full
Development of the Major Clusters, Supporting Clusters, Additional Clusters
and Mathematical Practices for this unit could include the following
instructional outcomes: FTF.A.1 ·
I can define a radian
measure of an angle as the length of the arc on the unit circle subtended by
the angle. ·
I can define terminal
and initial side of an angle on the unit circle. ·
I can explain how the
unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle. FTF.A.2 ·
I can explain the
relationship between the unit circle and the coordinate plane. FTF.A.3(+) ·
I can find the exact
value of trigonometric functions using the unit circle. ·
I can verify and
identify trigonometric identities. ·
I can use special
triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6. FTF.B.5 ·
I can define and
recognize the parameters of trigonometric functions. ·
I can interpret
trigonometric functions in realworld situations. ·
I can identify and
model periodic phenomena in realworld situations. ·
I can choose
trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline. FTF.C.8 ·
I can define
trigonometric ratios as related to the unit circle. ·
I can prove the
Pythagorean identity sin^{2}(Ө) + cos^{2}(Ө) =1. ·
I can use the Pythagorean identity, sin^{2}(Ө)
+ cos^{2}(Ө) =1, to find sin (Ө), cos (Ө), or tan
(Ө), given sin(Ө), cos(Ө), or tan(Ө), and the
quadrant of the angle. FTF.C.9(+) ·
I can verify and
identify trigonometric identities. ·
I can prove the addition
and subtractions formulas for sine, cosine, and tangent. ·
I can use the
addition and subtractions formulas for sine, cosine, and tangent to solve
problems. 

Enduring Understandings: ·
A radian measure of
an angle is the length of the arc on the unit circle subtended by the angle. ·
The unit circle
enables the extension of the domain of trigonometric functions to include all
real numbers. ·
Trigonometric
functions can be used to model periodic phenomena. ·
The Pythagorean
identity sin^{2}(θ) + cos^{2}(θ) = 1 is very useful when finding
sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. 
Essential Questions: ·
How can you find the
measure of an angle in radians? ·
What is the unit
circle? ·
How can you use the unit
circle to define the trigonometric functions of an angle? ·
What are the
characteristics of the reallife problems that can be modeled by
trigonometric functions? ·
How can you verify a
trigonometric identity? 
