Trigonometric Ratios 2013
Name: Tracy Harrison and Kelsey Skogsberg
Date: 12/11/2013
Grade Level: 9-10
Course: Honors Geometry
Time Allotted: 45 minutes
Number of students: 5 students
I.
Goal(s):
1. Students will be able to recognize which trigonometric ratio to use based on the
relative position of the reference angle and the given sides involved (adjacent and
opposite sides).
II.
Objective(s):
1. Students will analyze information from triangles.
2. Students will recognize previously used triangles such as isosceles triangles.
3. Students will apply sine, cosine, and tangent formulas to solve problems related to
triangles.
III.
Materials and Resources
1. Students will need iPads with the free Nearpod app.
2. (Optional) Students will need the print out of the PowerPoint to take notes on.
3. The teacher will need an iPad with the free Nearpod app, an additional computer,
and a projector.
IV.
Motivation (about 10 minutes)
1. The teacher will open the power point to the schedule for the class period.
2. The teacher will say, “Today, we are going to be working on the iPads on a Nearpod
lesson, so please, open the Nearpod app.
i. The pin number you will need for the lesson is shown on the top of the
teacher’s presentation.
 Write this up on the board for all students to refer to.
 Note: PIN changes depending on the session in order to send
specific assessment data to the Nearpod teacher for individual
lessons.
ii. The lesson that we are going to work on involves using trigonometric
ratios solve problems. We’re going to start by going over the today’s goal,
then reviewing what we have learned so far about trigonometric ratios.
iii. After we review, we’re going to do some practice problems and discuss
our different approaches.
iv. At the end of class, you will have a chance to give us some feedback about
how well this lesson helped you with trigonometric ratios, ask questions,
and we’ll end today with a learning log about what we did today in class.”
3. The teacher will ask for volunteers to read the goal listed on the first Nearpod slide.
i. “Students will develop strategies to recognize which trigonometric ratio to
use based on the relative position of the reference angle and the given
sides involved (adjacent and opposite sides).”
Trigonometric Ratios 2013
4. After reading the goal, students will be given a slide where they will have to label
the hypotenuse, reference angle, and adjacent and opposite sides of a triangle.
5. Once all the students have finished labeling the triangle, they will have a quiz asking
them to recall the ratios of sine, cosine, and tangent in terms of adjacent, opposite,
and hypotenuse.
i. After everyone finished the quiz, the teacher will pull up the solutions on
the power point and open the floor to questions.
Transition: “Now that we’ve briefly reviewed the material, let’s practice applying what we’ve
learned to different situations.”
V. Lesson Procedure (about 35 minutes)
(first section about 25 minutes)
6. The class will then work on solving 5 problems by first analyzing the information
provided, choosing a trigonometric ratio to use, and solving the problem leaving it in
radical or product form—except in special cases where the value of the ratio is
known.
i.
Problem 1
 Possible Solution
A. Sine
𝑥
sin(28°) = 19
19 sin(28°) = 𝑥
Trigonometric Ratios 2013
ii.
Problem 2
 Possible Solution
C. Tangent
8
tan(45°) = 𝑥
x[tan(45°)] = 8
8
𝑥 = tan(45°) = 8
iii.
Problem 3
 Possible Solution
B. Cosine
18
cos(61°) = 𝑥
𝑥[ cos(61°)] = 18
18
𝑥 = cos(61°)
Trigonometric Ratios 2013
iv.
Problem 4
 Possible Solution
A. Sine
6
sin(25°) = 𝑥
𝑥[sin(25°)] = 6
6
𝑥 = sin(25°)
v.
Problem 5
 Possible Solution
B. Cosine
6
cos(60°) = 𝑥
𝑥[cos(60°)] = 6
6
𝑥 = cos(60°) = 12
7. Once the class has gone through the problems, they will review the problems that
they just completed and look for shortcuts in solving the problems. The teacher
should ask questions such as, “Do we know any information about the given
degrees? Are any of these triangles special?” The goal is for students to recognize
that problem 2 and 5 can be used solving previous knowledge.
i.
For problem 2, they know that 45-45-90 triangles are isosceles triangles, and
thus, their legs should be equal in length.
Trigonometric Ratios 2013
ii.
Students may or may not recall that 30-60-90 triangles have special ratios for
cos(60°), but the teacher should remind them of a problem they did on a
previous day. (Note: In our class, they should have already worked with
special right triangles, and hopefully, they will recall this information.)
8. After the class discusses these shortcuts, they will be given a word problem about
rock climbing. They will have to draw a picture of the word problem and solve for
the length of rope.
i.
David and Emily are climbing El
Capitan, a big cliff wall in
Yosemite National Park. David
is on the ground holding the rope
attached to a karabiner (a rope
“pulley” that is on the wall)
above Emily as she climbs.
When Emily stops to rest, David
wonders how far away Emily will
be from him when she reaches the
karabiner.
The rope is attached to his waist
and then goes up to the karabiner and back down the wall to Emily’s
harness. The rope at David’s waist makes a 55° angle with the ground and he
is standing 20 feet away from the base of the wall.
Assuming that the rope is taut (i.e., pulled tight), approximately how long is
the rope between David and the karabiner above Emily?
 Possible Solution (Students may leave in as ratio)
(Second section about 10 minutes)
9. In the next section, students will be given two correlating word problems that focus
first on procedural knowledge of the step by step process of solving an equation and
then on conceptual knowledge of different approaches to problem solving. The
teacher can ask questions such as, “What do we see happening in our problem?
What do we know about the relationship of angles inside of a triangle?”
Trigonometric Ratios 2013
i.
Marta arrived for her geometry test only to find she forgot her calculator. She
decided to complete as much of each problem as possible.
She wrote & solved an equation. Her work is shown below. Explain what she did in each step.
29
sin(25°) = 𝑥
𝑥[sin(25°)] = 29
29
𝑥=
sin(25°)
ii.
 Possible solution
Step 1: She noticed that her reference angle was 65°; she was given the
opposite side, and she was looking for the hypotenuse. Thus, she set up
the sine ratio to solve the problem.
Step 2: She multiplied both sides by x.
Step 3: She divided both sides by sin(65°)
Ziv solved it differently. He used the equation to the below. Explain why
they both got the answer correct.
 Possible Solution
By the Triangle Angle Sum Theorem, we know that the angles inside of
the original problem triangle add up to 180°. Thus,
90 + 25 + other angle = 180
When we solve for the other angle, we get a value of 65°. Now that you
know the value of the missing angle, we can choose to use it as a
reference angle. By choosing to make 65° a reference angle, 29
becomes the adjacent side, and we are still looking for the hypotenuse.
Hence, we can step up an equation to solve for x the using cos(65°).
10. After students come to understand the conceptual knowledge of the second question,
they will be asked to apply this knowledge to solve a problem using two different
approaches and comparing their values. The teacher will tell the students that, when
Trigonometric Ratios 2013
they solve their problems until they need exact answers to compare, they may hit the
home button at the bottom of their iPad and open one of the graphing calculators to
calculate their solutions then return to Nearpod. In addition, teachers should remind
students to check their settings and set their angle measurement to degrees.
i.
Solve for y using sine and cosine.
 Solutions
Sine
𝑥
sin(53°) = 5
𝑥 = 5 sin(53°) ≈ 3.993
Cosine
90 + 53 = 143
180 – 143 = 37
𝑥
cos(37°) = 5
𝑥 = 5 cos(37°) ≈ 3.993
11. After completing this problem, students will be polled to see if they arrived at the
same solution. The class will briefly go over this solution and hypothesize that due
to the Triangle Angle Sum Theorem this works for all similar problems.
Transition: Now that you’ve had some practice working with trigonometric ratios, let’s see how
everyone is feeling on the material before we move on!
VI.
Closure (about 5 minutes)
12. To end the class period, students will be asked, “Based off of this lesson, how
comfortable do you feel identifying and applying trigonometric ratios?” They will
quickly choose one of the 4 responses depending on their comfort level.
13. In order to gain more formative feedback, the students will then answer the open
question, “What questions do you still have about trigonometric ratios or do you just
want more practice?”
14. Lastly, the students will complete a brief learning log responding to the prompt,
“Explain how you know which trigonometric ratio to use. Be sure to include
examples!” For examples, teachers should remind students of their new vocabulary
Trigonometric Ratios 2013
terms: reference angle, opposite side, adjacent side, hypotenuse, 45-45-90 triangle,
etc.
VII.
Extension (should take an additional 5 minutes)
1. The teacher will prompt the students to exit the first Nearpod presentation and enter
the PIN.
i. The pin number you will need for the lesson is shown on the top of the
teacher’s presentation
2. Then the teacher will ask for a volunteer to read the slide introducing the extension,
which briefly sums up that they have been using trigonometric ratios to find side
lengths and prompts the question, “What about finding angles?”
3. The students will then respond to a poll asking, “Can we find the degree of an angle
with the trig ratios we currently have?” After responses are made to the poll, there
will be a class discussion about their responses, and students will have to justify their
reasoning for their decision.
4. After the poll, students will be given some examples from which they can figure out
the angle because of the knowledge they have about the ratio of side lengths for 4545-90 and 30-60-90 special right triangles.
i. “From the ratio of side lengths, which angles can you find?”
5. After the student discussion about using the ratio of side lengths to work backwards
to find angles, students will be prompted with the open question, “How can you use
information about the side lengths to help you figure out the reference angle?”
Therefore setting the classroom up for the following lesson on inverse trigonometric
functions.
VIII.
Assessment
1. At the beginning of the lesson, the students go through a review in two parts (draw it
and quiz) to assess their current state of knowledge about sine, cosine, and tangent as
well as the correlating vocabulary (hypotenuse, reference angle, opposite side, and
adjacent side).
2. Throughout the lesson, students will be solving and discussing their solutions to
practice problems (procedural and conceptual problems)
Trigonometric Ratios 2013
3. At the end of the lesson, the students will have the opportunity to give feedback on
how they feel about their understanding of trigonometric ratios through a poll and an
open ended question where they will be able to ask questions or request more
practice problems.
4. The main summative assessment comes at the end of the lesson in the form of a
learning log where students will have to explain how they analyze the information
presented to them in order to choose the appropriate trigonometric ratio.
IX.
Standards
 Common Core State Standards for Mathematical Content
o CCSS.Math.Content.HSG-SRT.C.6: Understand that by similarity, side
ratios in right triangles are properties of the angles in the triangle, leading
to definitions of trigonometric ratios for acute angles. Explanation: The
students are continuing their exploration of sine, cosine, and tangent while
also introducing some new information that will lead to inverse functions.
o CCSS.Math.Content.HSG-SRT.C.7: Explain and use the relationship
between the sine and cosine of complementary angles. Explanation: For
problem 5-14, the discuss complementary angles and how cosine and sine
may be used in relation to those angles. It is also reviewed in the closure.
o CCSS.Math.Content.HSG-SRT.C.8: Use trigonometric ratios and the
Pythagorean Theorem to solve right triangles in applied problems.
Explanation: Throughout the lesson, they are using trigonometric ratios to
solve right triangle problems, and they specifically check the work of
another student who solved a trigonometric problem.
 Common Core State Standards for Mathematical Practice
o CCSS.Math.Practice.MP5: Use appropriate tools strategically.
Explanation: The students used sine, cosine, tangent, isosceles triangles,
and even, 30-60-90 special triangle rules to solve problems.
o CCSS.Math.Practice.MP8: Look for and express regularity in repeated
reasoning. Explanation: Students were asked to create and share strategies
for solving trigonometric problems with the date presented.