module 7 triangle trigonometry super final

GRADE 9 MATHEMATICSQUARTER 4Module 7

May 2014

Triangle TrigonometryMODULE 7

May 20, 2014

In a right triangle, one of the angles measures , and the remaining two angles are acute and complementary. The longest side of a right triangle is known as the hypotenuse and is opposite the right angle. The other two sides are called legs. The leg that is a side of an acute angle is called the side adjacent to the angle. The other leg is the side opposite the angle.

Lesson 1 The Six Trigonometric Ratios: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent

C

B

A

ac

b

Hypotenuse

Side Opposite

Side Adjacent

3


If two angles of a triangle are congruent to two angles of another triangle, the triangles are similar. If an acute angle of one right triangle is congruent to an acute angle of another right triangle, the triangles are similar, and the ratios of the corresponding sides are equal. Therefore, any two congruent angles of different right triangles will have equal ratios associated with them.

The ratios of the sides of the right triangles can be used to define the trigonometric ratios. The ratio of the side opposite and the hypotenuse is known as the sine. The ratio of the side adjacent and the hypotenuse is known as the cosine. The ratio of the side opposite and the side adjacent is known as the tangent.

4

SOH-CAH-TOA is a mnemonic device commonly used for remembering these ratios.

5


Words Symbol Definition

Trigonometric Ratios

sine sin

cosine cos

tangent tan

C

B

A

ac

b

Hypotenuse

Side Opposite

Side Adjacent

6


Words Symbol

Definition

Reciprocal Trigonometric Ratios

cosecant csc

secant sec

cotangent cot C

B

A

ac

b

Hypotenuse

Side Opposite

Side Adjacent

These definitions are called the reciprocal identities.

CHO-SHA-CAO is a mnemonic device commonly used for remembering these ratios.

In addition to the trigonometric ratios sine, cosine, and tangent, there are three other trigonometric ratios called cosecant, secant, and cotangent. These ratios are the reciprocals of sine, cosine, and tangent, respectively.

7

Lesson 2 Trigonometric Ratios of Special Angles

Consider the special relationships among the sides of - - and - - triangles.

x

x

2x60𝑜

30𝑜

y

y

y45𝑜

45𝑜

These special relationships can be used to determine the trigonometric ratios for , , and .

sin cos tan csc sec cot

Activity

Complete the table below that summarizes the values of the trigonometric ratios of the angles , , and .

8


x

x

2x60𝑜

30𝑜

y

y

y45𝑜

45𝑜

9

Answers:



2

1 1

2

10



2

1 1

2

Notice that = and = . This is an example showing that the sine and cosine are cofunctions. That is, if is an acute angle, sin = . Similar relationships hold true for the other trigonometric ratios.

11



2

1 1

2

sin = cos =

Cofunctions tan = cot =

sec = csc =

There are many applications that require trigonometric solutions. For example, surveyors use special instruments to find the measures of angles of elevation and angles of depression.

12

Lesson 3 Angles of Elevation and Angles of Depression

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An angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level.

14


An angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level.

15


The angle of elevation and the angle of depression are equal in measure because they are alternate interior angles.

16

Lesson 4 Word Problems Involving Right Triangles

Trigonometric functions can be used to solve word problems involving right triangles. The most common functions used are the sine, cosine, and tangent.

Moreover, you can use trigonometric functions and inverse relations to solve right triangles. To solve a triangle means to find all the measures of its sides and angles. Usually, two measures are given. Then you can find the remaining measures.

17


Example 1. A ladder is 12 feet long.

a) If the ladder is placed against a wall so that its base is 2 feet from the wall, find, to the nearest degree, the acute angle the ladder makes with the ground.

b) Suppose the base of the ladder is feet from the wall. Find an expression for , the angle the ladder makes with the ground.

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a) If the ladder is placed against a wall so that its base is 2 feet from the wall, find, to the nearest degree, the acute angle the ladder makes with the ground.

Solution.

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b) Suppose the base of the ladder is feet from the wall. Find an expression for , the angle the

ladder makes with the ground.

Solution.

20


Example 2. Latashi and Markashi are flying kites on a windy day. Latashi has released 250

feet of string, and Markashi has released 225 feet of string. The angle that Latashi’s kite string makes with the horizontal is . The angle that Markashi’s kite string makes with the horizontal is . Which kite is higher

and by how much?

21


Solution.

For Latashi’s kite:

Latashi’s kite has a height about 143.39 ft.

250 ft

35𝑜

Height = ?

22


Solution.

For Markashi’s kite:

Markashi’s kite has a height about 150.55 ft.

225 ft

42𝑜

Height = ?

23


Solution.

Let’s subtract the height of Markashi’s kite and the height of Latashi’s kite.

Markashi’s kite is higher than Latashi’s kite by about 7.16 ft.

24

Lesson 5 Oblique Triangles

Trigonometry enables sides and angle measures to be found in triangles other than right triangles.

An oblique triangle is one that does not contain a right angle.

Oblique triangles may be classifi ed into two---acute and obtuse.

An acute triangle is one that has three acute angles.

An obtuse triangle is one that has one obtuse angle.

25


Activity

Identify the acute and obtuse triangles.

26


Activity

Identify the acute and obtuse triangles.

27

Lesson 5.1 The Law of Sines and Its Applications

Law of Sines

Let be any triangle with , , and representing the measures of the sides opposite the angles with measures , , and , respectively.Then, the following are true.

28


From geometry, you know that a unique triangle can be formed if you know

a) the measures of two angles and the included side (ASA) or

b) the measures of two angles and the non-included side (AAS).

Therefore, there is one unique solution when you use the Law of Sines to solve a triangle given the measures two angles and one side.

29


From geometry, you know that

c) the measures of two sides and a non-included angle (SSA)

do not necessarily define a unique triangle. However, one of the following will be true.

1. No triangle exists.

2. Exactly one triangle exists.

3. Two triangles exist.

In other words, there may be no solution, one solution, or two solutions. A situation with two solutions is called the ambiguous case.

30

Lesson 5.1 The Law of Sines and Its ApplicationsSuppose you know the measures , , and . Consider the following cases.

31


ASA Case

Example 1. Cartography To draw a map, a cartographer needed to find the distances between point across the lake and each of point and on another side. The cartographer found miles, , and . Find the distances from to and from

to .

𝑋

𝑍

𝑌

32


ASA Example 1Solution.

The distance from to is about 0.59 miles.

𝑋

𝑍

𝑌 mi50𝑜100𝑜

33


ASA Example 1 (Continuation)Solution.

The distance from to is about 0.46 miles.

𝑋

𝑍

𝑌 mi50𝑜100𝑜

34


AAS Case

Example 2. A hill slopes upward at an angle of with the horizontal. A tree grows vertically on the hill.

When the angle of elevation of the sun is , the tree casts a shadow downhill that is 32 meters long. If the shadow is entirely on the hill, how tall is the tree?

35


AAS Case (Illustration)

Example 2. A hill slopes upward at an angle of with the horizontal. A tree grows vertically on the hill.

When the angle of elevation of the sun is , the tree casts a shadow downhill that is 32 meters long. If the shadow is entirely on the hill, how tall is the tree?

36


AAS Example 2Solution.a) Right Triangle Involving the Hill

Alone

b) Right Triangle Involving the Tree

c) Straight Angle Involving the Hill and Tree

d) Oblique Triangle Above the Hill

37


AAS Example 2 (Continuation)Solution.

The tree is about 15.6 meters tall.

38


SSA Case

Let’s have to cases.a) Case 1: b) Case 2:

Example 3. Determine the number of possible solutions for each triangle. a) , , b) , ,

39


SSA CaseExample 3. Determine the

number of possible solutions for each triangle.

a) , , Since , consider Case 1.

Since , there are two solutions for the triangle.

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SSA Case (Continuation)

Example 3. Determine the number of possible solutions for each triangle.

b) , , Since , consider Case 2.

In this triangle, , so there are no solutions.

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Lesson 5.1 The Law of Sines and Its ApplicationsSSA CaseOnce you have determined that there is/are one or two solution(s) for a triangle given the measures of two sides and a non-included angle, you can use the Law of Sines to solve the triangle.Example 4. Find all solutions for the triangle. If no

solutions exist, write none.

, ,

Since , consider Case 1.

Since , there are two solutions for the triangle.

42


Solution.Use the Law of Sines to find .

SSA Case Example 4 (Continuation)

Given:

, ,

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SSA CaseExample 4 (Continuation)

So, . Since we know there are two solutions, there must be another possible measurement for .

In the second case, must be less than and have the same sine value.

Since we know that if , , or is another possible measure for .

44


SSA Case Example 4 (Continuation)Now solve the triangle for each possible measure of .Solution I.

One solution is , , and .

A

B

C51𝑜 76.3𝑜

50

b

40

45


SSA Case Example 4 (Continuation)Solution II.

Another solution is , , and .

A

B

C

51𝑜

103.7𝑜

50

b

40

46


B

C

A

a

h

b

c

The area of any triangle can be expressed in terms of two sides of a triangle and the measure of the included angle.

Suppose you know the measures of and and the measure of the included angle in .

Let represent the measure of the area of , and let represent the measure of the altitude from . Then . But, or . If you substitute for , the result is the following formula.

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If you drew altitudes from A and C, you could also develop two similar formulas.

Area of Triangles

Let be any triangle with , , and representing the measures of the sides opposite the angles with measurements , , and , respectively. Then the area can be determined using one of the following formulas.

B

C

A

a

h

b

c

48


Example 5. Find the area of if , , and .

The area of is about 21.4 square units.

AB

C

4.7

12.447𝑜20′

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Deriving the Law of Cosines

• Write an equationusing Pythagorean theorem for shaded triangle.

b h a

k c - kA B

C

c

Abk

Abh

cos

sin

Abccba

AbccAAba

AbAbccAba

AbcAba

cos2

cos2cossin

coscos2sin

cossin

222

22222

222222

222

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Law of Cosines• Similarly

• Note the pattern

Cabbac

Baccab

Abccba

cos2

cos2

cos2

222

222

222

51

Lesson 5.2 The Law of Cosines and Its Applications

Law of Cosines

Let be any triangle with , , and representing the measures of the sides opposite the angles with measures , , and , respectively.Then, the following are true.

A

B C

b

a

c

Da - x

x

h

52


From geometry, you know that a unique triangle can be formed if

a) the measures of two sides and an included angle are known (SAS) or

b) the measures of three sides of a triangle are known and the sum of any two measures is greater than the remaining measure (SSS).

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SAS Case

Example 1. Landscaping Suppose you want to fence a triangular lot as shown at the right. If two sides measure 84 feet and 78 feet and the angle between the two sides is , what is the length of the fence to the nearest foot?

84 ft 78 ft

102𝑜

54


SAS Example 1Solution.

Let’s add the three lengths.

The length of the fence is about 288 ft.

84 ft 78 ft

102𝑜

55


SSS Case

Example 2. The sides of a triangle are 18 inches, 21 inches, and 14 inches. Find the measure of the angle with the greatest measure.

56


18 in21 in

14 in

SSS Case (Continuation)

Example 2. The sides of a triangle are 18 inches, 21 inches, and 14 inches. Find the measure of the angle with the greatest measure.

57


Formulas in Finding for the Angles

58


18 in21 in

14 in

with the greatest measure

SSS Example 2 (Continuation)Solution.

The largest angle has a measure about .

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References

Holliday, B. et al. (2004) Glencoe ADVANCED Mathematical Concepts . The McGraw-Hill Companies, Inc., United States of America

Senk, S. et al. (1998) UCSMP Functions, Statistics, and Trigonometry . Addison Wesley Longman, Inc., United States of America

Website Link for Video

http://www.youtube.com/watch?v=geDSwx2TuiE

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module 7 triangle trigonometry super final

Documents

angles of elevation

angles of depression

measures of angles

equal ratios

cosine cos cos

right angle

trigonometric solutions

angle of elevation