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YOUNGSTOWN CITY SCHOOLS
MATH: PRECALCULUS
UNIT 5: PROVING TRIGONOMETRIC IDENTITIES (2 WEEKS) 2013-2014
Synopsis: Students will be proving the Pythagorean identity and the addition and subtraction formulas for sine, cosine and
tangent. After the proofs have been presented, they will use these identities to solve problems.
STANDARDS
F.TF.8. Prove the Pythagorean identity sin2(θ) + cos2(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.
F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
MATH PRACTICES
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L-2 Communicate using correct mathematical terminology
L-3 Read, discuss, and apply the mathematics found in literature including looking at the author’s purpose
L-4 Listen to and critique peer explanations of reasoning
MOTIVATION
TEACHER
NOTES
1. Read poem on trigonometry, attached on page 8 or on the link listed below. Students then write their own
poems on trigonometry.
http://wiki.answers.com/Q/Can_you_help_you_write_an_acrostic_poem_for_trigonometry
2. The broadcasting example found on page 437 of the teacher’s textbook, Glencoe Advanced Mathematical
Concepts Precalculus with Applications is a good motivator. It explains interference in broadcasting and
goes on to solve it in example 4 on page 440.
3. Preview expectations for the end of the Unit
4. Have student set both personal and academic goals for this Unit.
TEACHING-LEARNING
Vocabulary
Sum identity for cosine
Difference identity for cosine
Pythagorean identity
TEACHER NOTES
Sum identify for sine
Difference identity for sine
Sum identity for tangent
Difference identify for tangent
1. Students need to know the Law of Cosines before they can prove the sum and difference identities. This
would also be a good time to present the Law of Sines since it seems to follow the material in Unit 6. So
use section 5-6 and 5-7 of the textbook and only discuss the Law of Sines and its ambiguous case.
Problems to look at are on page 316-317, numbers 11-18, 28, 31, 32, 33, 34; pages 324-325, numbers
11-37. The Law of Cosines is in section 5-8. Only discuss the Law of Cosines which would be examples
1 and 2. Problems to look at are on page 331-332, problems #11-18, 27, 31.
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TEACHING-LEARNING
TEACHER NOTES
2. Referring to a unit circle, discuss with students the basic identity:
and place them in
small groups to arrive at a proof for this identity. Possibly give them the hint to use the basic definitions
for
. Present students with several problems using this
identity. For example: given tan(x) = 12 and x is in the third quadrant, find sin(x) and cos(x). Ans:
3. Discuss the proof of the Pythagorean Identity,
, using the diagram on the web
site listed, if necessary: http://www.coolmath.com/lesson-pythagorean-identities-1.htm (F.TF.8, MP.1,
MP.2, MP.3, MP.4, MP.7, L.2)
4. Have students derive the two other forms of this identity, stating reasons for their steps:
and cos2(x) = 1 – sin2(x) (F.TF.8, MP.1, MP.2, MP.3, MP.4, MP.7, L,2, L,4)
5. After proving the three identities algebraically, take a graphical approach to the proofs. Examine the
functions formed by taking each side of the equation and graphing them to show that they are identical.
(F.TF.8, MP.1, MP.2, MP.4, MP.5, L,2)
6. Present to students the following problem or one similar to it: If cos x = and x is in the first quadrant,
find the value for the sin x using the Pythagorean identity. Pose the question: What would happen if x
is in the fourth quadrant? Why? Have students work several examples similar to the given example,
finding values for sin x, cos x, and tan x given one trig ratio and the quadrant of the angle. Discuss what
happens if the quadrant of the angle is not given. Worksheet #1 is included on page 9 for
reinforcement. (F.TF.8, MP.1, MP.2, MP.4, MP.7, MP.8, L,2)
7. Engage students in the proof of the difference identity for cosines on page 437 of the textbook. An
alternative proof of the sum and difference identities for cosine and sine are on the link:
http://www.milefoot.com/math/trig/22anglesumidentities.htm . Listed below are several ways to engage
students in proofs:
List the steps of the proof, leaving some of the steps out for the students to fill in
Write the identity that is to be proven and draw the picture on a piece of paper, hand students strips
of paper with the steps and reasons written on them and have them put the steps in order
underneath the identity and drawing
Write up the proof with errors in it and have students correct the errors.
(F.TF.9, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.4)
8. After students understand the proof of the difference identity for cosines, have them look at the proofs
for the sum identity for cosine (page 438), sum identity for sine (page 439), difference identity for
sine (page 439), sum identity for tangent and difference identity for tangent (problem #47 on page
444). (F.TF.9, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.4)
9. Students also need to be able to solve problems using these identities. The examples Attached on
pages 10-12 and those in your textbook on page 442-444 may help to explain the procedure used to
solve this type of problem. Work only those problems involving sine, cosine and tangent. (F.TF.9, MP.1,
MP.2, MP.4, MP.7, MP.8, L.2)
TRADITIONAL ASSESSMENT
1. Unit Tests: Multiple-Choice Questions
TEACHER NOTES
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TEACHER CLASSROOM ASSESSMENT
1. Teacher Classroom Assessments
2. Smaller authentic assessments as you go along
TEACHER NOTES
TEACHER
NOTES
AUTHENTIC ASSESSMENT (30% of grade)
1. Have students evaluate goals for the unit.
2. Students are to find errors and correct them in problems 1 thru 4 and disprove an identity by giving a
counterexample in problem 5 - - attached on page 4; Rubric and answers are on pages 5-7. (F.TF.8, F.TF.9,
MP.1, MP.2, MP.3, MP.7, MP.8, L.2)
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AUTHENTIC ASSESSMENT PROBLEMS:
1. The Pythagorean identity states:
Adding a cos2 (x) from each side we have
By taking the square root of each side we have
2. Find the tan (x-y) given 0< x<
3. Simplify: cos (x
and
sin2 (x) = 1 – cos2 (x)
sin (x) = 1 – cos (x)
< y< π, sin x =
)
cos (x
)=
cos x * cos + sin x * sin
cos x * 1 + sin x * 0 =
cos x
=
4. Find the exact value for
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5. Prove: sin y * tan y = cos y is not a trig identity by using a counterexample
AUTHENTIC ASSESSMENT RUBRIC
ELEMENTS OF THE
PROJECT
Pythagorean identity
0
1
2
3
Did not
attempt
Corrected one
error, no
explanation
Corrected both
errors, no explanation
Corrected both errors
and gave explanation
Difference identity for Did not
tangent
attempt
Corrected one
error, no
explanation
Corrected both
errors, no explanation
Corrected both errors
and gave explanation
Difference identity for Did not
cosine
attempt
Corrected one
error, no
explanation
Corrected both
errors, no explanation
Corrected both errors
and gave explanation
Sum identity for sine
Did not
attempt
Corrected one
error, no
explanation
Corrected both
errors, no explanation
Corrected both errors
and gave explanation
Disprove identity
with counterexample
Did not
attempt
Chose a value for y
such that the
identity was true
Chose a value for y
which disproved the
identity, errors in the
proof
Chose a value for y
which disproved the
identity with no errors
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Answers for the Authentic Assessment:
1. The Pythagorean identity states:
Adding a cos2 (x) from each side we have
sin2 (x) = 1 – cos2 (x)
By taking the square root of each side we have
sin (x) = 1 – cos (x)
2
In the second line you are subtracting a cos (x) from each side.
When you take the square root of each side you get sin (x) =
root of each term.
2. Find the tan (x-y) given 0< x<
and
< y< π, sin x =
Explanation: tan x =
and the tan y =
quadrant making it negative.
3. Simplify: cos (x
, you cannot take the square
since tangent is defined as
and y is in the second
)
cos (x
)=
cos x * cos + sin x * sin
cos x * 0 + sin x * 1 =
sin x
When looking at the unit circle, cos
=
, sin
4. Find the exact value for
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When looking at a 45-45-90 triangle on the unit circle, sin
5. Prove: sin y * tan y = cos y is not a trig identity by using a counterexample.
Students can choose a value for y, preferably one for which they know the sine, cosine and
tangent and one that disproves the identity. Say you choose
cos . Which means
for y, then the identity becomes sin
* tan
=
which is false so the identity does not work.
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TWO POEMS ON TRIGONOMETRY
Tremendous labor is what it takes
Rigorous worksheets when you start to accomplish
Identities, not our own, but with six trigonometric functions
Graphing sine cosine tangent along with their inverses
Operations are still set with equalities and inequalities
No other way but the mastery of the basic tools in circles
Onward with complex, quadratics and its applications
Meted with exponential and logarithmic functions
Endeavor does not end because we'll meet Gauss-Jordan and Cramer
To submerge with determinants, matrices - even augmented
Ready or not the FUN will surely start
Yes but with PATIENCE though bitter we'll surely PASS!
[email protected]
Time may give you pressure,
Relax and let your mind do the work,
In Trigo, it's all the complex equations,
Give your best shot and solve them all,
Over and over, repeat the solutions,
Nothing should stop you, must concentrate,
On a table with a paper and pencil,
Must answer your best before time runs out,
Erase your mistakes but learn from it,
Try again, don't give up,
Remember the lessons, reflect on them,
You can do it, it's just a subject with numbers.
~Sarleen Castro
Westfields International School, Philippines
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T/L #6: WORKSHEET #1
Using the Pythagorean identity, solve the following problems:
1. Sin θ =
and 180< θ<270, find the cos θ and tan θ
2. Tan θ =
and 180< θ<270, find the cos θ and sin θ
3. Cos θ =
and 270< θ<360, find the sin θ and tan θ
4. Tan θ =
and
5. Cos θ =
, find the sin θ and cos θ
, find sin θ and tan θ in terms of x
6. Sin θ =
find cos θ and tan θ
7. Cos θ =
, find sin θ and tan θ
8. Tan θ =
, find sin θ and cos θ
_____________________________________________________________________________________________
Answers:
1. cos θ =
2. cos θ = -½ and sin θ=
3. sin θ =
and tan θ =
4. sin θ =
5. sin θ =
and tan θ =
6. cos = ½ and tan
7. sin θ =
8. sin θ =
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T-L #9 Problems:
Example 1
a. Show by producing a counterexample that cos (x - y) ≠ cos x - cos y.
b. Show that the difference identity for cosine is true for the values used in part a.
Answers:


a. Let x = 0 and y = 4. First find cos (x - y) for x = 0 and y = 4.


cos (x - y) = cos (0 - 4)
Replace x with 0 and y with 4.


= cos (-4)

0 - 4 = -4
2
= 2
Now find cos x - cos y.


cos x - cos y = cos 0 - cos 4
Replace x with 0 and y with 4.
2 2- 2
= 1 - 2 or 2
So, cos (x - y) ≠ cos x - cos y.

b. Show that cos (x - y) = cos x cos y + sin x sin y for x = 0 and y = 4.
2

First find cos (x - y). From part a, we know that cos (0 - 4) = 2 .
Now find cos x cos y + sin x sin y.


cos x cos y + sin x sin y = cos 0 cos 4 + sin 0 sin 4
= (1)
Substitute for x and y.
( 22) + (0) ( 22)
2
= 2

Thus, the difference identity for cosine is true for x = 0 and y = 4.
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Example 2
Use the sum or difference identity for cosine to find the exact value of cos 825°.
Answer:
825° = 2(360°) + 105°
Symmetry identity, Case 1
cos 825° = cos 105°
cos 105° = cos (60° + 45°)
= cos 60° cos 45° - sin 60° sin 45°
60° and 45° are two common angles whose sum is 105°.
Sum identity for cosine
1 2
3 2
=2 2 - 2  2
2- 6
4
=
Therefore, cos 825° =
2- 6
4 .
Example 3
12
15


Find the value of sin (x + y) if 0 < x < 2, 0 < y < 2, sin x = 13 and sin y = 17.
Answer:
In order to use the sum identity for sine, we need to know cos x and cos y. We can use a Pythagorean
identity to determine the necessary values.
sin2 α + cos2 α = 1 so cos2 α = 1 - sin2 α Pythagorean identity
Since we are given that the angles in Quadrant I, the values of sine and cosine are positive. Therefore,
cos α = 1 - sin2 α.
cos x =
=
cos y =
 
1  12
13
2
25
5
169 or 13
 
1  15
17
2
64
8
or
289 17
Now substitute these values in to the difference identity for sine.
sin (x + y) = sin x cos y + cos x sin y
12 8
5 15
= 13 17 + 13 17
=
( )( ) ( )( )
171
= 221 or about 0.774
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Example 4

TRIGONOMETRY If the sine wave is shifted 2 units
to the left, it coincides with the cosine wave. Therefore,

cos  = sin ( + 2 ) is an identity. Prove the identity.
Answer:

cos  = sin ( + 2)


cos  = sin  cos 2 + cos  sin 2
cos  = sin  (0) + cos  (1)
cos  = cos 
The identity is true.
Example 5
Use the sum or difference identity for tangent to find the exact value of tan 345°.
tan 345° = tan (300° + 45°)
300° and 45° are two common angles whose sum is 345°.
tan 300° + tan 45°
= 1 - tan 300° tan 45°
Sum identity for tangent
=
tan 300° = - 3, tan 45° = 1
=
Multiply by
1- 3
to simplify.
1- 3
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