Sara Lasker
10/21/12
Learning Plan 1
York High School
Trigonometry Lesson Plan
Title: Sum & Difference and Double-Angle Identities
Content Area: Trigonometry
Grade Level: 10th/11th/12th grade
Time Frame: 1 period (100 minutes)
Date: November 2012
SOL: T.5 The student will verify basic trigonometric identities and make substitutions using the
basic identities.
Related SOLs: T.1 The student, given a point other than the origin on the terminal side of an
angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent,
cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions
defined on the unit circle will be related to trigonometric functions defined in right triangles.
Lesson Objectives:
 The students will use the sum and difference identities, along with the special angles, to
calculate exact values for the sine and cosine of the sum or difference of two angles.
 The students will, given a single trig function value of each of two angles, use right
triangle trig tools to determine the sine and cosine of the angles. They will then use the
appropriate identities to calculate the sine and cosine of the sum or difference of the
given angles.
 The students will use the double-angle identities, along with the special angles, to
calculate exact values for the sine and cosine of twice a given angle.
 The students will, given a single trig function value to an angle, use right triangle trig
tools to determine the sine and cosine of the angle. They will then use the appropriate
identities to calculate the desired sum.
Materials/Resources:
 Student-generated identity page
 Unit circle
 Classwork sheet
 Worksheet for homework
Instructional Procedures:
(1) Warm-Up: All students complete two review problems from the last lesson at their seats.
Go over warm-up once students are done. (15 minutes)
(2) Introduce lesson with cos(𝐴 − 𝐵) example. (5 minutes)
(3) Introduce the sum and difference identities for sine, cosine, and tangent. Students will
write these identities on the identity page they’ve been making throughout the semester.
(10 minutes)
(4) Work examples on board for students to copy into their notebooks for reference. (15
minutes)
(5) Practice: Students get into pairs and go to dry erase boards to complete problems using
sum and difference identities. Each partner must participate. Walk around room
checking steps and solutions and answering questions. Students may not move on to the
next question until their answer has been checked. The questions will be projected on the
front board. (20 minutes)
(6) Students return to their desks. Introduce the double-angle identities for sine, cosine, and
tangent. Students will write these identities on their identity pages. Have the students
show that the double-angle identities are a simplification of the sum identities when the
two angles are the same. (10 minutes)
(7) Work examples on board for students to copy into their notebooks for reference. (10
minutes)
(8) Practice: Pass out classwork. Students complete problems at their seats using sum and
difference and double-angle identities. They may work individually or with those sitting
around them. Walk around to check work and answer questions. This will be collected
for a classwork grade. Anything not finished in class will be completed for homework.
(15 minutes)
(9) Hand out review worksheet for students to complete for homework. This will be due
next class.
Independent Procedures:
(1) Choose a partner for dry erase activity.
(2) Choose to work individually or with others on practice problems.
Homework: Worksheet
Assessment:
(1) Give feedback to each group of students during the dry erase activity
(2) Monitor students’ work and results during individual class work. This will be collected,
and credit will be awarded based on completion and showing all the steps.
(3) Students will complete graded worksheet for homework.
Extension Activities/Interdisciplinary Links: N/A
Differentiation Strategies/Activities: Modify assignment and assessment for students with
IEPs. The student with emotional behavior disorder may work alone on the dry erase activity if
she prefers. The autistic boy will complete every other problem in part two of the homework and
will be given an extra day to finish that and the classwork.
Assessment (Formal): Homework, classwork
Assessment (Informal): Observation, classwork
Resources (Text, Technology, Other): N/A
Extension Activities (Interdisciplinary Links): N/A
Bloom’s Taxonomy: Knowledge, Comprehension, Application
Marzano’s Nine High Yield Strategies: Reinforcing Effort & Providing Recognition,
Homework & Practice, Cooperative Learning, Setting Objectives & Providing Feedback
Warm-Up
Simplify each expression.
1.
2.
tan 𝑥 csc 𝑥
sec 𝑥
1
tan2 𝑥
+ tan 𝑥 cot 𝑥
3. Prove that tan 𝑥 + cot 𝑥 = sec 𝑥 csc 𝑥.
Intro to Sum & Difference (p. 215)
𝜋
Does cos(𝐴 − 𝐵) = cos 𝐴 − cos 𝐵? Let A = and B = 0.
2
Intro to Double-Angle (p. 233)
Let B = A in the identity cos(𝐴 + 𝐵).
Sum and Difference Identities
sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
tan 𝐴 + tan 𝐵
tan(𝐴 + 𝐵) =
1 − tan 𝐴 tan 𝐵
tan 𝐴 − tan 𝐵
tan(𝐴 − 𝐵) =
1 + tan 𝐴 tan 𝐵
Double-Angle Identities
sin 2𝐴 = 2 sin 𝐴 cos 𝐴
cos 2𝐴 = cos 2 𝐴 − sin2 𝐴
cos 2𝐴 = 1 − 2sin2 𝐴
cos 2𝐴 = 2 cos 2 𝐴 − 1
2tan 𝐴
tan 2𝐴 =
1 − tan2 𝐴
Name: _________________________
Date: _____________
Block: _______
Sum & Difference Identities
1. Find the exact value of sin 75°.
13𝜋
11𝜋
− tan
12
12
13𝜋
11𝜋
1+tan
tan
12
12
tan
2. Write
as a single angle. Then find the value of the angle.
Double Angle Identities
Write as a single angle. Then find the value of the angle.
𝜋
𝜋
1. 2sin 3 cos 3
2.
2tan 67.5°
1−tan2 67.5°
Classwork (Dry Erase Boards)
5𝜋
1. Find the exact value of cos 12 .
tan 160°+tan 140°
2. Write 1−tan 160° tan 140° as a single angle. Then find the value of the angle.
3. Verify the equation cos(𝑥 + 𝑦) + cos(𝑥 − 𝑦) = 2 cos 𝑥 cos 𝑦.
Name: ________________________
Date: ____________
Block: ______
Classwork: Sum & Difference and Double-Angle Identities
Write as a single angle. Then find the value of the angle.
1. sin 15° cos 15°
𝜋
2. cos 3 cos
3.
4𝜋
3
𝜋
− sin 3 sin
4𝜋
3
tan 30°
2(1−tan2 30°)
Write as a single angle. Then find the value of the angle, and evaluate the remaining five trig
functions.
4. sin 75° cos 15° − cos 75° sin 15°
Verify the following equation.
5. sin 2𝑥(tan 𝑥 + cot 𝑥) = 2
Name: ________________________
Date: ____________
Block: ______
Homework: Sum & Difference and Double-Angle Identities
1. Use a sum or difference formula to evaluate:
a. sin 15°
b. cos 255°
2. Write as a single angle, then find the value of the angle:
11𝜋
7𝜋
11𝜋
7𝜋
a. sin 12 cos 12 − cos 12 sin 12
b. cos 2 22.5° − sin2 22.5°
𝜋
c. 2cos2 12 − 1
𝜋
𝜋
e. 2sin 8 cos 8
g.
tan 5°+tan 40°
1−tan 5° tan 40°
3. Prove: sin(𝜋 + 𝑥) = − sin 𝑥
d. cos 150° cos 30° − sin 150° sin 30°
f. sin 10° cos 20° + cos 10° sin 20°
7𝜋
7𝜋
h. sin2 12 − cos2 12