NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
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Lesson 27: Sine and Cosine of Complementary Angles and
Special Angles
Student Outcomes
Students understand that if and are the measurements of complementary angles, then .
Students solve triangle problems using special angles.
Lesson Notes
Students examine the sine and cosine relationship more closely and find that the sine and
cosine of complementary angles are equal. Students become familiar with the values
associated with sine and cosine of special angles. Once familiar with these common
values, students use them to find unknown values in problems.
Classwork
Example 1 (8 minutes)
Students discover why cosine has the prefix “co-”. It may be necessary to remind students
why we know alpha and beta are complementary.
Example 1
If and are the measurements of complementary angles, then we are going to show that
.
In right triangle , the measurement of acute angle is denoted by , and the measurement
of acute angle is denoted by .
Determine the following values in the table:
What can you conclude from the results?
Since the ratios for and are the same, and ratios for and
are the same; additionally, . The sine of an angle is equal to the cosine
of its complementary angle, and the cosine of an angle is equal to the sine of its
complementary angle.
Scaffolding:
If students are struggling
to see the connection, use
a right triangle with side
lengths 3, 4, and 5 to help
make the values of the
ratios more apparent.
Use the cutouts from
Lesson 21.
Ask students to calculate
values of sine and cosine
for the acute angles (by
measuring) and then ask
them, "What do you
notice?"
As an extension, ask
students to write a letter
to a middle school student
explaining why the sine of
an angle is equal to the
cosine of its
complementary angle.
MP.2
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
415
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Therefore, we conclude for complementary angles and that or, in other words, when
that
and . Any two angles that are complementary can
be realized as the acute angles in a right triangle. Hence, the “co-” prefix in cosine is a reference to the fact that
the cosine of an angle is the sine of its complement.
Exercises 1–3 (7 minutes)
Students apply what they know about the sine and cosine of complementary angles to solve for unknown angle values.
Exercises 1–3
1. Consider the right triangle so that is a right angle, and the degree measures of and are and ,
respectively.
a. Find .
b. Use trigonometric ratios to describe
two different ways.
,
c. Use trigonometric ratios to describe
two different ways.
,
d. What can you conclude about and ?
e. What can you conclude about and ?
2. Find values for that make each statement true.
a.
b.
c.
d.
3. For what angle measurement must sine and cosine have the same value? Explain how you know.
Sine and cosine have the same value for . The sine of an angle is equal to the cosine of its complement. Since
the complement of is , .
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
416
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Example 2 (8 minutes)
Students begin to examine special angles associated with sine and cosine, starting with the angle measurements of
and . Consider modeling this on the board by drawing a sketch of the following figure and using a meter stick to
represent .
Example 2
What is happening to and as changes? What happens to and
?
There are values for sine and cosine commonly known for certain angle measurements. Two such angle
measurements are when and .
To better understand sine and cosine values, imagine a right triangle whose hypotenuse has a fixed length of
unit. We illustrate this by imagining the hypotenuse as the radius of a circle, as in the image.
What happens to the value of the sine ratio as approaches ? Consider what is happening to the opposite
side, .
With one end of the meter stick fixed at , rotate it like the hands of a clock and show how decreases as decreases.
Demonstrate the change in the triangle for each case.
As gets closer to , decreases. Since
, the value of is also approaching .
Similarly, what happens to the value of the cosine ratio as approaches ? Consider what is happening to the
adjacent side, .
As gets closer to , increases and becomes closer to . Since
, the value of is
approaching .
Now, consider what happens to the value of the sine ratio as approaches Consider what is happening to
the opposite side, .
As gets closer to , increases and becomes closer to . Since
, the value of is also
approaching .
What happens to the value of the cosine ratio as approaches ? Consider what is happening to the
adjacent side, .
As gets closer to , decreases and becomes closer to . Since
, the value of is
approaching .
Remember, because there are no right triangles with an acute angle of or of , in the above thought
experiment, we are really defining and .
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
417
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Similarly, we define and ; notice that this falls in line with our conclusion that the sine of
an angle is equal to the cosine of its complementary angle.
Example 3 (10 minutes)
Students examine the remaining special angles associated with sine and cosine in Example 3. Consider assigning parts
(b) and (c) to two halves of the class and having students present a share out of their findings.
Example 3
There are certain special angles where it is possible to give the exact value of sine and cosine. These are the angles that
measure , , , , and ; these angle measures are frequently seen.
You should memorize the sine and cosine of these angles with quick recall just as you did your arithmetic facts.
a. Learn the following sine and cosine values of the key angle measurements.
Sine
Cosine
We focus on an easy way to remember the entries in the table. What do you notice about the table values?
The entries for cosine are the same as the entries for sine but in the reverse order.
This is easily explained because the pairs , , and are the measures of complementary angles. So,
for instance, .
The sequence
may be easier to remember as the sequence
.
b. is equilateral, with side length ; is the midpoint of side . Label all side lengths and angle
measurements for . Use your figure to determine the sine and cosine of and .
Provide students with a hint, if necessary, by suggesting they construct the angle bisector of , which is also the
altitude to .
MP.7
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
418
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,
,
,
c. Draw an isosceles right triangle with legs of length . What are the measures of the acute angles of the triangle?
What is the length of the hypotenuse? Use your triangle to determine sine and cosine of the acute angles.
,
Parts (b) and (c) demonstrate how the sine and cosine values of the mentioned special angles can be
found. These triangles are common to trigonometry; we refer to the triangle in part (b) as a ––
triangle and the triangle in part (c) as a –– triangle.
Remind students that the values of the sine and cosine ratios of triangles similar to
each of these will be the same.
Highlight the length ratios for –– and –– triangles. Consider using a set up like
the table below to begin the conversation. Ask students to determine side lengths of three
different triangles similar to each of the triangles provided above. Remind them that the scale
factor will determine side length. Then, have them generalize the length relationships.
–– Triangle,
side length ratio
–– Triangle,
side length ratio
Scaffolding:
For the
triangle,
students may develop the
misconception that the
last value is the length of
the hypotenuse; the
longest side of the right
triangle. Help students
correct this misconception
by comparing
and
to show that
,
and
, so
.
The ratio
is easier
to remember because of
the numbers .
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
419
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Exercises 4–5 (5 minutes)
4. Find the missing side lengths in the triangle.
,
,
5. Find the missing side lengths in the triangle.
,
,
Closing (2 minutes)
Ask students to respond to these questions about the key ideas of the lesson independently in writing, with a partner, or
as a class.
What is remarkable about the sine and cosine of a pair of angles that are complementary?
The sine of an angle is equal to the cosine of its complementary angle, and the cosine of an angle is
equal to the sine of its complementary angle.
Why is ? Similarly, why is , , and ?
We can see that approaches as approaches . The same is true for the other sine and cosine
values for and .
What do you notice about the sine and cosine of the following special angle values?
The entries for cosine are the same as the entries for sine, but values are in reverse order. This is
explained by the fact the special angles can be paired up as complements, and we already know that
the sine and cosine values of complementary angles are equal.
Sine
Cosine
Exit Ticket (5 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
420
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 27: Sine and Cosine of Complementary Angles and Special
Angles
Exit Ticket
1. Find the values for that make each statement true.
a.
b.
2. is a –– right triangle. Find the unknown lengths and .
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
421
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Exit Ticket Sample Solutions
1. Find the values for that make each statement true.
a.
b.
2. Triangle is a –– right triangle. Find the unknown lengths and .
Problem Set Sample Solutions
1. Find the value of that makes each statement true.
a.
b.
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
422
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c.
d.
2.
a. Make a prediction about how the sum will relate to the sum .
Answers will vary; however, some students may believe that the sums will be equal. This is explored in
problems (3) through (5).
b. Use the sine and cosine values of special angles to find the sum: .
and
; therefore,
.
Alternative strategy:
c. Find the sum: .
and
; therefore,
.
d. Was your prediction a valid prediction? Explain why or why not.
Answers will vary.
3. Langdon thinks that the sum is equal to . Do you agree with Langdon? Explain what this
means about the sum of the sines of angles.
I disagree. Explanations may vary. It was shown in the solution to Problem 3 that , and it is
known that
. This shows that the sum of the sines of angles is not equal to the sine of the sum of
the angles.
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Lesson 27
GEOMETRY
Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Date: 10/28/14
423
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
4. A square has side lengths of
. Use sine or cosine to find the length of the diagonal of the square. Confirm your
answer using the Pythagorean theorem.
The diagonal of a square cuts the square into two congruent
–– right triangles. Let represent the length of the
diagonal of the square:
Confirmation using Pythagorean theorem:
5. Given an equilateral triangle with sides of length , find the length of the altitude. Confirm your answer using the
Pythagorean theorem.
An altitude drawn within an equilateral triangle cuts the
equilateral triangle into two congruent –– right
triangles. Let represent the length of the altitude:
The altitude of the triangle has a length of
.
Confirmation using Pythagorean Theorem:
