Name: ___________________________ Learning Module 27

Gr.& Section: _____________________

FOURTH QUARTER

“PROBLEM SOLVING
INVOLVING THE SIX TRIGONOMETRIC RATIOS”

The following topics are included in this module:

1) Applications Involving Right Triangles

In this module, you are expected to meet the following competencies:

1) Use trigonometric ratios to solve real – life problems involving right triangles
2) Solve real-life problems involving right triangles and circular functions with precision and accuracy.

Pre–Test
Before we proceed with the lesson, let us first test your prior knowledge about the topic in this module.
Directions: Translate the following into a sketch as accurately as possible. Then, label the drawings.
1. 10 ft inclined plane elevated at 17o
2. 35 ft pole with a shadow of 72 ft
3. Buoy at an angle of depression of 59o from lighthouse top 175 ft high
4. Tree top elevated 29o from a point 50 ft from the base
5. 200 ft up a 10o slope
Refer to the Answer Key on the last page of this module to check your answers.

Day 1: Applications Involving Right Triangles (Synchronous Session)

CONCEPT NOTES
In this section, the solution of a right triangle as a tool in solving real-life problems will be the focus.

ILLUSTRATION

1. Find the height of a pine tree, if 47 ft from its base the angle of elevation of its top 35o.
Given:
θ (angle of elevation) = 35°
adjacent side (distance between A and C) = 47 ft
Find:
Opposite side (Height of a pine tree)
Formula:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
Tan A =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
Let x be the length of opposite side
Solution:
𝑥
Tan 35° =
47
x = (tan 35°)(47)
x = 32.91 ft
Therefore, the height of a pine tree is 32.91 ft

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 2. An observer standing on the top of a vertical cliff 51 ft tall spots a boat in the adjacent valley. If the
base of the cliff is 60 ft from the boat, find the angle of depression from an observer.
Given:
Opposite side (distance between B and C) = 51 ft
Adjacent side (distance between A and B) = 60 ft
Find:
θ (angle of depression)
Formula:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 51 𝑓𝑡
Tan θ = =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 60 𝑓𝑡
Solution:
Tan θ = 0.85 = 400
θ = 40o
Therefore, the angle of depression is 40o
For more examples, refer to your book “Grade 9 Mathematics Patterns and Practicalities” on page 448-
450.
EXERCISE NO. 27.1
(NOTE: Write your answers on a Learning Activity Sheet.)
Directions: Answer “Try This” on page 448 and page 450 (Grade 9 Mathematics Patterns and
Practicalities).

Day 2: Activity Involving Right Triangles (Synchronous Session)

EXERCISE NO. 27.2
(NOTE: Write your answers on a Learning Activity Sheet.)
Directions: Turn to page 455 of your book Grade 9 Mathematics Patterns and Practicalities and answer
“Problem Solving” item numbers 25-26.

Day 3: Asynchronous Activity

Activity no. 27.1
(NOTE: Write your answers on a Learning Activity Sheet.)
Time started:
Time finished:
Directions: Read the question carefully and write a reflection about it in at least five (5) sentences. Be
guided by the provided criteria. Write your answers on a separate sheet of paper.
“Look at situations from all angles, and you will become more open.”
1. What does this quotation mean?
2. Why do you think you need to look at situations from all angles?

Day 4:(Synchronous Session)

EXERCISE NO. 27.3

Directions: Turn to page 455 of your book Grade 9 Mathematics Patterns and Practicalities and answer
“Problem Solving” item numbers 27-30.
*Please note that the answers for this activity will be presented in class.

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Day 5: Post-Test (Synchronous Session)
Post–Test
Let us test what you have learned after accomplishing this module. Read the directions carefully and
answer the test below.

UNDERSTANDING
Directions: Read and answer the question concisely. Refer to the rubric as your guide. Write your answer on
the space provided.
Criteria 4 points 3 points 2 points 1 point
Uses clear, consistent Uses clear and Uses clear organizational Uses inconsistent
organizational strategy: consistent strategy but with organizational strategy:
Organization
presentation is logical. organizational occasional presentation is not
strategy. inconsistencies. logical.
The answer is focused, The answer is The answer is focused on The answer poorly
purposeful, and reflects focused on the topic topic and includes few addresses topic and
Content
original insight and ideas. and includes relevant loosely related ideas. includes irrelevant
ideas. ideas.
1. As they say, when the “going gets tough, the tough gets going.” What does this mean? How do you cope
when things get difficult or challenging?
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You are done with Module 27. Congratulations! You made it!
For clarifications, do not hesitate to consult your teacher.

Reference
(1) Gladys C. Nievera Ph.D. Grade 10 Mathematics Patterns and Practicalities.

Answer Key
Here are the answers in pre-test. Please look at this part only after accomplishing the pre-test.
HONESTY IS STILL THE BEST POLICY. Please check your answers and determine how much
knowledge you have attained.
Pretest

1. 2. 3. 4. 5.

59o
10ft 35ft 175ft 200ft
o
17
29o 10o
72ft 50ft

How many items did you get correctly?

Get ready to learn more in the next learning module!

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means without the prior written permission of the author.