Jocelyn Straight
We don’t have to turn in lesson plans at DGS so I don’t have a format to follow!
The following Lesson would utilize technology by using the Smartboard
Illinois Learning Standards for Mathematics:
6.A.4 Identify and apply the associative, commutative, distributive and identity properties of
real numbers, including special numbers such as pi and square roots.
6.B.3b Apply primes, factors, divisors, multiples, common factors and common multiples in
solving problems.
6.B.3c Identify and apply properties of real numbers including pi, squares, and square roots.
8.C.5 Use polynomial, exponential, logarithmic and trigonometric functions to model situations.
9.D.4 Analyze and solve problems involving triangles (e.g., distances which cannot be measured
directly) using trigonometric ratios.
9.D.5 Analyze and solve problems involving periodic patterns (e.g., sound waves, tide variations)
using circular functions and communicate results orally and in writing.
These standards represent some of the goals that this lesson plan hopes to achieve. The goals
that are given in the Illinois Learning Standards for Mathematics are very vague and seem to
be non-inclusive of many of the topics that are covered in an Advanced Algebra II and Trig
class (AAT 300). The only mention of proof based reasoning seems to be in relation to
geometry proofs.
Content Area
1. Simplifying and Verifying Trigonometric Identities
2. Proof (concentrating on 2-column)
3. Algebraic properties (e.g. distribution prop, commutative prop, substitution, combining
like terms)
Currently in our AAT 300 classes we have our students simplify and verify trig identities
without stating the reason for each of their steps. Since the majority of our students taking
AAT 300 have just finished taking a year of Geometry, they are well acquainted with 2-column
proofs. The students will need to know how to set up a 2-column proof (prior knowledge), know
when to correctly use algebraic properties, and know the trig identities they just learned
(reciprocal identities, tangent and cotangent identities, and the Pythagorean identities)
Objectives
1. To improve students abilities to justify the steps that they take while simplifying and
verifying trigonometric identities
2. To improve students abilities to help the “reader” follow the logic taken during their
solutions to problems dealing with simplifying and verifying trigonometric identities
3. To improve students abilities to follow the correct algebraic steps needed to reach their
solutions
4. To help students see the connections between the algebraic “rules” that we use for
Algebra I problems and how they are applied to AAT 300 topics
Sequence of Activities and Learning Activities
1. I will start class with a basic proof that all students had seen in their Geometry classes
from last year. I will choose a problem that is geometry based, but relies on algebra
mainly to solve
a. Given: AB = CD
A
B
C
D
Prove: AC = DB
2. I will have the students work independently on the problem as I walk around the room
3. I will select different students to show the class their work. (selection will be based on
the different ways, both correct and incorrect that students have shown) If students
don’t remember how to formally prove this problem, they may try to use specific values
as “proof”. I will explain what constitutes a proof and how showing examples, no matter
how many, don’t constitute a valid proof.
4. After going through the different ways that students approached this problem and
answering any questions we will discuss how proof has a place in the algebra classroom as
well.
5. We will then practice writing 2-column proofs when solving algebraic equation
a. Given: 9(x – 5) + 7 = 7x + 50
Prove: x = 44
9( x  5)  7  7 x  50
given
9 x  45  7  7 x  50
distributive property
9 x  38  7 x  50
combine like terms
2 x  38  50
subtraction property
2 x  88
addition propetry
x  44
division propetry
6. I will first see how the students approach an algebraic proof. We will discuss what
constitutes a statement and reason. Many students may not think there are reasons for
algebraic steps (since we don’t rely on postulates and theorems the way that geometry
does). As we go through the proof I will ask them “why” they did the step that they did.
For example, they may say that they distributed, and then I would say that we just used
the distributive property and that’s our reason.
7. We will then make a list of reasons that we can use for an algebraic proof, much like the
ones that geometry student’s use.
8. After the students have compiled a list of reasons, along with the identities that we have
just learned dealing with trig functions, we will start simplifying and verifying trig
identities. For example:
a.
Prove: cos 2 x  tan 2 x cos 2 x  1
cos 2 x  tan 2 x cos 2 x
given
sin 2 x
cos 2 x
2
cos x
2
cos x  sin 2 x
cos 2 x 
1
substitution
Simplifying
substitution
b. Simplify: sec x cot x  cot x cos x
sec x cot x  cot x cos x
given
1 cos x cos x cos x

substitution
cos x sin x sin x 1
1
cos 2 x

simplifying
sin x sin x
1  cos 2 x
combine like terms
sin x
sin 2 x
substitution
sin x
sin x
simplifying
9. We will continue doing trig proofs in this manner. This will force the students to think
about the steps that they are taking while it will encourage them to show each step
10. Their HW will ask them to verify or simplify trig identities using the 2-column format.
11. We will go over the HW the following day by having students share their answers
Example problems for more in-class practice and independent practice
Simplify the following trig expressions completely
1.
15.
2.
16.
3.
17.
4.
18.
5.
19.
6.
20.
7.
8.
21.
22.
23.
9.
24.
10.
25.
11.
26.
12.
27.
13.
14.
AAT 300
Section 14.3 Day 2
28.
Name_____________________________
More Trig Identity Practice
DO ALL WORK ON A SEPARATE SHEET OF PAPER
Prove the following identities:
Worksheet B (1 – 15)
Worksheet C (16 - 30)
1.
sec x(sec x  cos x)  tan x
16.
tan x(cot x  tan x)  sec2 x
2.
sin x(csc x  sin x)  cos 2 x
17.
cos x(sec x  cos x)  sin 2 x
3.
csc2 x  cos 2 x csc2 x  1
18.
cos 2 x  tan 2 x cos 2 x  1
4.
(sec  1)(sec  1)  tan 2 x
19.
(1  sin x)(1  sin x)  cos 2 x
5.
sec2 x  tan 2 x sec2 x  sec4 x
20.
cot 2 x csc2 x  cot 2 x  cot 4 x
6.
cos 4 x  sin 4 x  1  2sin 2 x
21.
sec4 x  tan 4 x  1  2 tan 2 x
7.
1
cos x

 tan x
sin x cos x sin x
sin x cos x

1
csc x sec x
1
 csc 2 x  csc x cot x
1  cos x
cos x
cos x

 cot 2 x
2
sec x  1 tan x
sec x
 sec 2 x  sec x tan x
sec x  tan x
22.
sec x sin x

 cot x
sin x cos x
1
1

1
2
sec x csc 2 x
1
 sec 2 x  sec x tan x
1  sin x
sin x
1  cos x

 2 csc x
1  cos x
sin x
1  sin x
 2sec 2 x  2sec x tan x  1
1  sin x
12.
sin 3 x cos 2 x  cos 2 x sin x  cos 4 x sin x
27.
sin 3 x cos 2 x  sin 3 x  sin 5 x
13.
sec2 x  csc2 x  sec2 x csc2 x
28.
sec x  tan x 
14.
1  3cos x  4cos 2 x 1  4cos x

sin 2 x
1  cos x
29.
sec2 x  6 tan x  7 tan x  4

sec2 x  5
tan x  2
15.
sin 3 x  cos3 x
 1  sin x cos x
sin x  cos x
30.
sec3 x  cos3 x
 sec2 x  1  cos 2 x
sec x  cos x
8.
9.
10.
11.
2
23.
24.
25.
26.
1
sec x  tan x