REPUBLIC OF THE PHILIPPINES DEPARTMENT OF EDUCATION LEARNING GUIDE

FOURTH YEAR - MATHEMATICS

CIRCULAR FUNCTIONS AND TRIANGLE TRIGONOMETRY

LAWS ON TRIANGLE TRIGONOMETRY MADE SIMPLE

impose as a condition the payment of royalties.

Mind Map

The Mind Map displays the organization and relationship between the concepts and activities in this Learning Guide in a visual form. It is included to provide visual clues on the structure of the guide and to provide an opportunity for you, the teacher, to reorganize the guide to suit your particular context.

Stages of Learning

The following stages have been identified as optimal in this unit. It should be noted that the stages do not represent individual lessons. Rather, they are a series of stages over one or more lessons and indicate the suggested steps in the development of the targeted competencies and in the achievement of the stated objectives.

Assessment

All six Stages of Learning in this Learning Guide may include some advice on possible formative assessment ideas to assist you in determining the effectiveness of that stage on student learning. It can also provide information about whether the learning goals set for that stage have been achieved. Where possible, and if needed, you can use the formative assessment tasks for summative assessment purposes i.e as measures of student performance. It is important that your students know what they will be assessed on.

1. Activating Prior Learning

This stage aims to engage or focus the learners by asking them to call to mind what they know about the topic and connect it with their past learning. Activities could involve making personal connections.

Background or purpose

The prior learning of the students on triangles needs to be activated in this stage. They shall state the relations among the sides and the angles in a triangle.

Strategy

BUZZ SESSIONS. This strategy aims to maximize students' engagement and to tap their

Materials

 activity sheets on pages 20-21  masking tape

 manila paper  protractor

 pentel pen  pair of scissors

 centimeter grid on page 22

Activity 1: “Laws to Generate”

Instructions:

1. Organize the class into groups of 5 or as desired. 2. Distribute the materials needed to each group.

3. Encourage them to work cooperatively on the task and derive the correct answers. 4. Ask them to finalize their answers on a manila paper to be presented to the class.

Formative Assessment

Monitor the involvement of the students in the group activity. Check their outputs.

Roundup

The students should have stated the relations among the sides and the angles in a triangle.

2. Setting the Context

This stage introduces the students to what will happen in the lessons. The teacher sets the objectives/expectations for the learning experience and an overview how the learning experience will fit into the larger scheme.

Background or purpose

In this stage, the students will solve problems involving the measures of angles or sides in a right triangle using the trigonometric functions. They will also be challenged to solve a problem on oblique triangle using trigonometric functions, if applicable.

Strategy

TIP (Think-Ink-Pair). A strategy that allows individual to reach consensus and check

understanding. Students think individually about the problem and write their own understanding. After which, everyone is given a chance to discuss with a partner and record what they have drawn to reach consensus.

Materials

 manila paper  masking tape

 pentel pen

Activity 2: “Laws to Comprehend”

Instructions:

2. Ask each student to look for a partner and share his/her answers to reach a consensus. 3. Then, write the following problems on a manila paper for the students to solve. Problem 1: Find b.

Problem 2: Find a.

Below are the possible answers to the problems. 1. cos Ө= adjacent

hypotenuse cos 190 ₌ b

70 b = 70(cos 190₎ b = (70)(0.9455) b ≈ 66.19

2. sin α = _hypotenuseopposite sin 400 ₌ a

39cm a = 39cm (sin 400₎ a = 39(0.6428) a ≈ 25.07 cm

1. Is the figure a right triangle? Support your answer.

2. Can you still make use any of the six trigonometric functions to solve the lengths of sides b and c?

 If yes, show your answer.

 If no, then this will be discussed in the succeeding stage.

Formative Assessment

Check the answers of the students on the problems.

Roundup

The students should have solved the problems involving the measures of angles or sides of a triangle using the trigonometric functions.

3. Learning Activity Sequence

This stage provides the information about the topic and the activities for the students. Students should be encouraged to discover their own information.

Background or purpose

In this stage, different engaging activities are suggested to carry out the concepts that will provide opportunities for the students to demonstrate ability in solving problems on triangles applying the trigonometric functions, and laws of sines and cosines.

Strategies

 TIP (Think-Ink-Pair). A strategy that allows individual to reach consensus and check

understanding. Students think individually about the problem and write their own

understanding. After which, everyone is given a chance to discuss with a partner and record what they have drawn to reach consensus.

 INTERACTIVE LECTURE. This strategy provides students with a general outline to give them

a framework for thinking about a subject and to structure their notetaking. This type of lecture involves students by focusing their attention on key concepts. It emphasizes information transfer at the knowledge, recall, and comprehension levels of learning.

 TRIVIA GAME. A baffling game that requires an accurate response. This aims to develop

triangles using laws of sines and cosines. This strategy asserts the proficiency of the students on the topic being discussed.

Materials

 illustrations drawn on the manila papers  Teacher Resource Sheet 1 on page 23  Student Activity 4 sheets on pages 27-28

 manila paper  pentel pen  masking tape

Introductory Activity and Teacher's Input

Begin this stage by presenting the problem below to the class. Allow them to understand and solve this on their seats in a couple of minutes and then call for volunteers to show their solutions on the board.

Questions:

1. What is asked in the problem?

2. Draw on the illustration the angle made by the ladder with the ground. 3. How will you solve the problem?

4. What do you think is the correct answer?

Among the six trigonometric functions, the equation that best describes the problem is sinӨ= opposite

hypotenuse.

Solutions:

sinӨ=opp hyp

Trigonometric function described in the problem.

sin 56o ₌ 10 cm x

Substitute the given information to the equation.

x sin 56o _{= 10 cm Multiply both sides of the equation by x.} x sin 560

sin 560 = 10 cm sin 560

Divide both sides of the resulting equation by sin 56o_.

x = 10 cm 0.8290

Compute the value of sin 56o _{using a calculator.Press}

or to get . Then, get the quotient.

x = 12.06 cm Answer to the problem.

The study of trigonometric functions is originated with the geometric problems involving triangles. Solving triangles means finding the lengths of the sides and measures of the angles of the triangle. Problems on triangles involve the angles of elevation and depression.

Now, present to the class the following definitions and illustrations of the angles of elevation and depression.

sin 5 6 5 6 sin

A figure illustrating the angle of elevation The angle of elevation of the object from the observer is αΟ_.

The angle of depression of the object from the observer below is βΟ_.

A figure illustrating the angle of depression

You may discuss to them the illustrative examples below. When possible, ask them first if they can solve the problems on their own.

Problem # 1:

As you can see, this type of problem requires a diagram of a carefully labeled right triangle. The measurements of the sides and one angle should be labeled using the information given. One measurement in the triangle is missing. It is the goal of the problem to find this measurement. First, you should assign a variable to represent the missing measurement. We usually use a lower case English letter e.g. x, y, etc. to represent the measure of a side of the triangle and either a Greek letter such as α for alpha, β for beta, Ө for theta, and so on or an upper case English letter to represent a missing angle measurement.

A trigonometric ratio often helps us set up an equation which can then be solved for the missing measurement. If the two legs of the triangle are part of the problem, then it is a tangent ratio. If the hypotenuse is part of the problem, then it is either sine or cosine ratio. To illustrate the problem above with the given the information, we have,

Solutions:

tan α = _adjacentopposite tan 41o ₌ 50 ft

x x tan 41o _{= 50 ft.}

41O 50ft.

x tan 410

tan 410 =

50 ft. tan 410

x

=

Problem # 2:

An airplane is flying at a height of 4 kilometers above the ground. The distance along the ground from the airplane to the airport is 6 kilometers. What is the angle of depression from the airplane to the airport?

Solutions:

tan α = _adjacentopposite tan α = 4 km_{6 km} tan α = 0.6667

α = tan-1_(0.6667)

α≈ 33.69O

The solution may be obtained also using a calculator or a table of trigonometric ratios. Using a

scientific calculator, press to display .

Some scientific calculators do not have an key, instead they have .

But still, you can derive at the same answer using any of these keys. Some scientific calculators don't have either of these keys, rather a .

To derive the same display using this key, press .

Therefore, the angle of depression from the airplane to the airport is approximately 33.69o_.

. 6 6 6 7 INV tan 39.69

INV 2ndF

SHIFT

Activity 3: “Laws to Incorporate”

Instructions:

1. Organize the class into groups of 5 or as desired.

2. Prepare 2 to 3 sets of each problem found on page 23 in order to have 2 to 3 groups solving a common task for comparison of answers.

3. Distribute to each group the materials needed. 4. Let them work cooperatively the given task.

5. Ask them to finalize their answers on a manila paper for the presentation of outputs. 6. Check their answers. The possible solutions are found on page 24.

In the previous stage, you have posed the questions below for the students to think.

✔ Is the figure a right triangle? Support your answer.

✔ Can you still make use any of the six trigonometric functions to solve the lengths of sides

b and c?

The measure of ∠B can easily be obtained by subtracting the sum of 43O _{and 57}O _{from 180}O_{, thus} 180O _{– (43}O _{+ 57}O_{) = 80}O_{. But how about the lengths of sides b and c?}

We have been able to use the trigonometric functions to solve right triangles. With the six trigonometric ratios, we were able to solve the measures of the angles or sides of the right triangles.

There are identities used to find the unknown measures of the angles or sides of oblique triangles and two of these are the laws of sines and cosines.

LAW OF SI NES

LAW OF SI NES

The Law of Sines is fairly easy to follow and very useful to solve a triangle when the following information will be given:

SSA two sides and the angle not included between them,

SAA two angles and one side that is not included in the angles.

Legend: S – Side; A - Angle

The law of sines is described by the given relation, _sinAa

=

=

Illustrative Example 1:

Now, let's make use of the previous figure and solve the missing lengths of the sides b and c of

∆ABC using the law of sines.

Given: a = 4, ∠A = 43O_{, and ∠C = 57}O_. Solutions:

Since, side b and ∠B are not yet given, we can use the formula, _{sin A}a = c

sin C from a

sin A= b sin B=

c sin C. a

sin A = c sin C

Formula to use to solve first the value of c.

4 sin 43O=

c sin 57O

Substitute the given information to the equation.

csin 43O

 =4sin 57O

 Cross-multiply the equation.

c0.6820 =40.8387 Compute the value of sin 43o _{and sin 57}o _{using a}

calculator.

0.6820c=3.3548 Simplify the resulting equation.

c=3.3548 0.6820

c=4.92

To solve for the length of side b, use the formula, a sin A =

b

sinB. You noticed that ∠B is

unknown, but take note that the sum of the angles of a triangle is 180o_{, therefore we can get the} measure of ∠B by subtracting the sum of 43O _{and 57}O _{from 180}O_{. So, 180}O _{– (43}O _{+ 57}O_{) = 80}O_.

a sin A =

b sinB

b0.6820=40.9848 0.6820b=3.9392

4 sin 43O=

b

sin80O b=

3.9392 0.6820 bsin 43O = 4sin80O b=5.78

LAW OF COSI NES

_{LAW OF COSI NES}

Oblique triangles can also be solved using another identities which is called the Law of Cosines. This law states that “in any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides subtracting two times the product of the lengths of these sides and the cosine of the included angle.”

There are three rules that make up the statement of the law, but we could only need to memorize one because the other two can be obtained by changing the letters (put b in place of a, for example). All three rules are listed below.

The law of cosines can be used when the following information will be given: SAS two sides and an included angle, and

SSS the lengths of all three sides.

Legend: S – Side; A - Angle Illustrative Example 2:

In ∆ABC, a = 24, c = 32, and m∠B = 115O_{. Solve for the measures of the other side and angles of} the triangle.

Solutions:

Since two sides and an included angle are given, we can use the law of cosines to find the third side and the measures of the other two angles.

b2 _{= a}2 _{+ c}2 _{– 2ac(cos B)}

b2 _{= (24)}2 _{+ (32)}2 _{– [2(24)(32)][cos 115}O₎ b2 _{= 576 + 1024 – 2(768)(-0.4226)} b2 _{= 1600 + 649.11}

b2 _{= 2249.11}



₌



To solve for ∠A, use the rule, a2 _{= b}2 _{+ c}2 _{– 2bc(cos A).} a2 _{= b}2 _{+ c}2 _{– 2bc(cos A)}

(24)2 _{= (47.4)}2 _{+ (32)}2 _{– [2(47.4)(32)][cos A]} 576 = 2246.76 + 1024 – [(3033.6)(cos A)] 576 = 3270.76 - [(3033.6)(cos A)] 576 – 3270.76 = -3033.6 cos A -2694.76 = -3033.6 cos A

−

−

=

−

−

cos A = 0.8883 A = cos-1 _(0.8883) A = 27.3O

To solve for the measure of ∠C: m∠C = 180O _{- (m∠A + m∠B)}

= 180O _{– (27.3}O _{+ 115}O₎ = 180O _{– 142.3}O m∠C = 37.7O

Activity 4: “Laws to Verify”

Instructions:

1. Let the same groups of students work this activity.

2. Distribute to each group the Student Activity 4, Law to Verify, worksheets on pages 27-28.

3. Ask them to complete the activity in a given time allotment you set.

4. Let them post their outputs for comparison and discussion. Refer to page 29 for the answer key.

Formative Assessment

Roam around and check if the students are doing the given tasks correctly and to ensure that everyone in each group has contributed their ideas from the different activities.

Checking of groups' outputs should be done every after the activity.

Roundup

The students should have demonstrated their abilities in solving problems on right triangles using trigonometric functions and problems involving triangles using the laws of sines and cosines.

4. Check for Understanding of the Topic or Skill

This stage is for teachers to find out how much students have understood before they apply it to other learning experiences.

Background or purpose

The activity suggested in this stage aims to ensure and check how far the students have gained knowledge on solving problems on triangles using trigonometric functions and the laws of sines and cosines.

Strategy

DECODING. A strategy used to translate data or message from a code into the original language or form. In the context of this activity, the students will solve problems on triangles using trigonometric functions and the laws of sines and cosines. After which, students will look for the corresponding answers on the decoder that will satisfy the given challenge.

Material

• activity sheets on pages 30-32

Activity 5: “Laws to Validate”

Instructions:

1. Reorganize the class into groups of 4 or as desired. 2. Distribute to each group the activity sheets.

3. Set a time allotment for them to complete the activity.

4. Let them post their outputs when they will be through for comparison and discussion.

Formative Assessment

Check their outputs. Refer to page 33 for the answer key.

Roundup

The students should have solved problems on triangles using the trigonometric functions and the laws of sines and cosines. They were able to consolidate and verify also the key ideas on triangle trigonometry in the given activity.

5. Practice and Application

In this stage, students consolidate their learning through independent or guided practice and transfer their learning to new or different situations.

Background or purpose

In this stage, the students will explore some real-life applications of the trigonometric functions and the laws of sines and cosines by solving word problems. Through this suggested activity, they will appreciate the importance of the study of triangle trigonometry.

Strategies

 TASK CARDS. A strategy which encourages small groups of students to work for a common

goal. Specific task is identified in a card for them to complete and later discuss their outputs in a whole group. This activity enhances their mathematical and interpersonal intelligences.

 PROBLEM SOLVING. Teaching students how to effectively solve problems will provide them

with useful lifelong skills. Problem solving models, such as working mathematically model, break problem solving into a step by step process:

CLARIFY What is the problem asking you to do or find out? What is given in the problem?

CHOOSE What tools would be effective for solving the problem? USE Use the tools to gain an answer to the problem.

INTERPRET Is this answer reasonable? Can you check it using another method?

Materials

 task cards on pages 34-36  manila paper

 pentel pen  masking tape

Activity 6: “Laws to Apply”

Instructions:

1. Organize the class into six (6) groups.

2. Assign in every group a leader to facilitate the given tasks.

3. Make sure that you have prepared two sets of each activity sheet in order to have two groups solving the common tasks.

5. Let them finalize their answers on a manila paper for the presentation of outputs. 6. Ask a representative from each group to discuss their solutions to the class.

Formative Assessment

Check the outputs of the students using the answer key on pages 37-38.

Roundup

The students were able to explore some real-life applications of the trigonometric functions and the laws of sines and cosines by solving word problems. With this, it is hoped that they have appreciated the importance of the study of triangle trigonometry.

6. Closure

This stage brings the series of lessons to a formal conclusion. Teachers may refocus the objectives and summarize the learning gained. Teachers can also foreshadow the next set of learning

experiences and make the relevant links.

Background or purpose

This stage aims to consolidate the learning gained by the students on the concepts introduced in this module by writing journals on the things they learned, found interesting and experienced difficulty in understanding the triangle trigonometry.

Strategies

 JOURNALS. This strategy provides a good opportunity for students to reflect on their

learning. Journals provide a good source of assessment for teachers. Sometimes, these may not be written – diagrams and other drawings are also authentic forms of demonstrating learning.

 REFLECTIVE WRITING USING 3-2-1. This is a strategy that is very effective in assessing

students' understanding of the concepts. In this context, the students will be asked to write 3 things they learned about the lesson, 2 things they find interesting, and 1 thing they want to learn more about it. This strategy will give you, as teacher, an idea on how you effectively carried out the lesson.

Material

• reflective journal on page 39

Activity 7: “Laws to Consolidate on My Reflective Journal”

Instructions:

1. Let the students write in their journal individually based from the concepts they learned and how those key ideas are useful in their daily experiences.

2. Hand out to them the reflective journal sheet.

3. Encourage them to highlight the concepts they learned, find interesting and want to learn more.

4. Instruct them to get ready for the sharing of journals as soon as they finish writing.

Formative Assessment

Emphasize to the students that a lot of the succeeding lessons will make use of their understanding about the topics they have undertaken.

Roundup

The students should have consolidated their learning gained on the concepts introduced in this module by writing journals on the things they learned, found interesting and experienced difficulty in understanding the triangle trigonometry. It is important to address those things they want to learn more by conducting a remediation of the lesson.

Teacher Evaluation

(To be completed by the teacher using this Teacher’s Guide) The ways I will evaluate the success of my teaching this unit are: 1.

STUDENT ACTIVITY 1

LAWS TO GENERATE

State the relations among the sides and the angles in a triangle.

Directions:

Read all the instructions to make important discoveries about triangles by measuring and taping strips of paper together.

1. Cut out strips from the centimeter grid paper provided in your group.

2. Fold and tape both ends of the strip together that can possibly form a triangle. For instance, a strip with measurements of 5 cm, 5 cm and 5 cm would be cut and folded as follows.

3. Use the measurements below to possibly form triangles. Be sure to label each strip with its letter.

A - 6 cm, 6 cm, 6 cm

B - 4 cm, 9 cm, 5 cm

C - 7 cm, 4 cm, 7 cm

D - 4 cm, 4 cm, 8 cm

E - 5 cm, 5 cm, 8 cm

4. Using a protractor, get the measures of the angles in each triangle, if possible. A ______ ______ ______

B ______ ______ ______ C ______ ______ ______ D ______ ______ ______ E ______ ______ ______ F ______ ______ ______

Questions:

1. Were you able to form triangles using the given measurements? Why? 2. Can any three lengths be the sides of a triangle? Why or why not?

3. Compare the sum of the lengths of any two sides of a triangle with its third side. What discoveries do you have about the lengths of the sides in a triangle?

4. Is your finding true to all the given measurements in item #3? Support your answer.

STUDENT ACTIVITY 1

CENTIMETER GRID PAPER

TEACHER RESOURCE SHEET 1

LAWS TO INCORPORATE

LAW TO INCORPORATE

POSSIBLE SOLUTIONS

1.

tan β = _adjacentopposite

tan 35o ₌ 25m

x x tan 35O _{= 25 m}

x tan 35O tan 35O =

25 m tan 35O

x = 25 m 0.7002

x = 35.7 meters

2.

tan α = _adjacentopposite

tan 33o ₌ x

100 x = 100 (tan 33o₎

x = 100 (0.6494)

x = 64.94 feet

3.

tan α = _adjacentopposite

tan α = 22.5m 34m tan α = 0.6618

α ≈ tan-1 _(0.6618)

TEACHER RESOURCE SHEET 2

DERIVATIONS OF THE LAWS OF SINES AND COSINES

THE LAW OF SINES

Consider an oblique triangle (triangle with no right angle) with side a opposite angle A,

side b opposite angle B, and side c opposite angle C. Draw perpendicular segments from the vertex at Cto its opposite side and from the vertex at Ato its opposite side and call them h and h' respectively.

With this, the following formulas can be determined.

sin A = h b hence, h = b sin A

sin B = h a h = a sin B

a sin B= b sin A Equate the expressions for h.

a sin B sin A sin B=

b sin A sin A sin B

Divide both sides by sin

α

β

a sin A =

b sin B

Simplify.

Using h', b sin B=

c

sin C can be shown. By transitivity,

a sin A =

b sin B =

c sin C.

REMEMBER:

In any oblique triangle, a side divided by the sine of the angle opposite it, is equal to any other side divided by the sine of the opposite angle.

THE LAW OF COSINES

Consider a triangle on a coordinate plane such that the vertex of angle A is at origin and side c lies along the positive x axis.

If u and v denote the lengths as indicated in the figure below, the coordinates of the vertex at C are (u, v).

If cos A= adj

hyp = u

b, then u = b cos A. Similarly, if sin A= opp hyp=

v

b, then v = b sin A. Thus, the coordinates of the vertex at C are (b cos A, b sin A).

Using the distance formula,

d

=





x

−

x





y

−

y



d=



=



−2bc cos Ac2

b2_sin2_A



=



cos2_A

sin2_A

−2bc cos Ac2



=



Squaring both sides results in a2 _{= b}2 _{+ c}2 _{– 2bc cos A.}

The other two equations are derived in the same way.

STUDENT ACTIVITY 4: LAWS TO VERIFY – WORKSHEET 1

Directions: Use the laws of sines or cosines to solve the given problems involving triangles. Locate your answers in the grid on the next page, then cross out the letters. Read the remaining letters from right to left to get the answer of the given puzzle.

1. In ∆ABC, m∠A = 35O_{; m∠B = 15}O_{; c = 5; find}

the measure of ∠C and the lengths of sides a and b.

2. In the figure, m∠A = 30O_{; b = 4 and c = 2.}

Find the measures of ∠B, ∠C and the length of side a.

3. Use the given information to solve the

triangle below. 4. Given the three sides of a triangle, find all the angles.

5. In ∆ABC, a = 4.56, m∠A = 43O_{, and m∠C =}

57O_{. Solve the triangle.}

6. Use the data in the figure below to solve the triangle. A B C a b c A B C a b c A B C a = 6

b

c 100O

30O

A

B

C a = 5

b = 3 c = 6

A

B

C a = 4.56

b c A B C 120O

a = 10

STUDENT ACTIVITY 4

LAWS TO VERIFY – WORKSHEET 2

M A R K R A N F

c = 5.61 a = 8.12 _∠B = 89.9O _{∠C = 50}O _{∠B = 79.3}O _{b = 13.8 c = 7.1 b = 1.69}

O R S U I C I D

∠B = 80O _{a = 2.48 ∠A = 77.1}O _{b = 14.6 c = 9.19 A = 30.2 ∠A = 100.3}O _{B = 15.23}

E I N H I S U P

∠C = 22.7O _{∠C = 23.9}O _{b = 6 a = 3.74 ∠A = 90}O _{∠B = 21.4}O _{∠C = 110}O _{a = 3.11}

P E R A N D C O

∠C = 130O _{b = 11.82 c = 8.78 c = 5.19 b = 6.58 ∠B = 29.9}O _{∠A = 66.4}O _{∠C = 83}O

R R E C T P A T H

∠B=126.1O _{a = 5.04 a = 3.33 c = 4.46 b = 18 c = 9.82 ∠B = 37.3}O _{∠A = 56.3}O _{∠C = 93.8}O

STUDENT ACTIVITY 4

LAWS TO VERIFY

ANSWER KEY

WORKSHEET 1

No. 1

∠

C = 130

a = 3.74

b = 1.69

No. 2

∠

B = 126.7

∠

C = 23.3

a = 2.48

No. 3

∠

C = 50

c = 9.19

b = 11.82

No. 4

∠

A = 56.3

∠

B = 29.9

∠

C = 93.8

No. 5

∠

B = 80

c = 5.61

b = 6.58

No. 6

∠

B = 37.3

∠

C = 22.7

c = 4.46

WORKSHEET 2

Answer:

PTEROCARPUS INDICUS - NARRA

M

A

R

K

R

A

N

F

O

R

S

U

I

C

I

D

E

I

N

H

I

STUDENT ACTIVITY 5

LAWS TO VALIDATE

Objective: Demonstrate ability to solve problems on triangles using trigonometric functions and the laws of sines and cosines.

DECOD E R

Triangle trigonometry can be used to solve problems involving the angles of

and

. However, many interesting problems involve non-right triangles.

The identities needed to solve these type of problems are the

STUDENT ACTIVITY 5

LAWS TO VALIDATE

ANSWER KEY

A

opp

hyp

I

15.69 ft

_R

21.35 ft

C

25.07

_L

60

S

157.87 ft

D

adj

hyp

N

72.97

_T

_{33.59 ft}

E

8

_O

39.8

_V

_96.41

F

55.3

P

opp

adj

W

27.18

STUDENT ACTIVITY 6

LAWS TO APPLY

Directions

: Give what is asked in each item below. Show your solutions.

Problem 1:

Solve the problem given its illustrations.

Problem 2: Illustrate and solve.

A vertical flagpole is attached to the top edge of a building. A man stands

200 feet from the base of the building. From his viewpoint, the angle of

elevation to the bottom of the flagpole is 60

_{; to the top is 62.5}

_{. Determine}

STUDENT ACTIVITY 6

LAWS TO APPLY

Directions

: Give what is asked in each item below. Show your solutions.

Problem 1:

Solve the problem given its illustration.

Problem 2: Illustrate and solve.

Two of the angles in a triangle are 30

_{and 45}

_{. If the side opposite the 45}

angle has length of

20

STUDENT ACTIVITY 6

LAWS TO APPLY

Directions

: Give what is asked in each item below. Show your solutions.

Problem 1:

Solve the problem given its illustration.

Problem 2:

Illustrate and solve

.

STUDENT ACTIVITY 6

LAWS TO APPLY

ANSWER KEY

TASK 1

adj

= 6 m 2 m = 3 x = tan-1₍₃₎

x ≈ 71.57O

2. tan 60O ₌ opp

adj

let n = height of the building

tan 60O ₌ n

200 ft n = 200(tan 60O₎

= 200(1.7321)

≈ 346.42 ft

tan 62.5O ₌ 346.42x

200

346.42 + x = 200(tan 62.5O₎

346.42 + x = 384.2

x = 384.2 – 346.42

x ≈ 37.78 ft.

TASK 2

adj

let n = height of the building

tan 60O ₌ n

200 ft n = 200(tan 60O₎

= 200(1.7321)

≈ 346.42 ft

tan 62.5O ₌ 346.42x

200

346.42 + x = 200(tan 62.5O₎

346.42 + x = 384.2

x = 384.2 – 346.42

x ≈ 37.78 ft.

2. 20 7 sin 45o=

x sin 30o

x(sin 45O_{) =} 20

7 sin 30

o



x(0.7071) = 1.4286

x = 1.4286 0.7071

TASK 3

adj

tan 35O ₌ x

15 m x = 15(tan 35O₎

= 15 (0.7002)

≈ 10.5

Height of a kite:

10.5 + 1.5 = 12 m

2. sin 60O ₌ opp

hyp

sin 60O ₌ x

10 ft x = 10(sin 60O₎

= 10(0.8660)

STUDENT ACTIVITY 7

LAWS TO CONSOLIDATE

MY REFLECTIVE JOURNAL

For the Teacher: Translate the information in this Learning Guide into the following matrix to help you prepare your lesson plans.

Stage

_1.

Activating Prior

Learning

2.

Setting the

Context

3.

Learning

Activity Sequence

4.

Check for

Understanding

5.

Practice and

Application

6.

Closure

Strategies

Activities from the Learning Guide

Extra activities you may wish to include

Materials and planning needed

Estimated time for this Stage