trigonometry! ;)

8/8/2019 Trigonometry! ;)

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Submitted by:

Gianne Denise C. Dueas

IV-Vanadium


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Trigonometry is the study and solution of

Triangles. Solving a triangle means findingthe value of each of its sides and angles. The

following terminology and tactics will be

important in the solving of triangles.

Pythagorean Theorem (a2+b2=c2). Only for right angle triangles

Sine (sin), Cosecant (csc or sin-1)

Cosine (cos), Secant (sec or cos-1)

Tangent (tan), Cotangent (cot or tan-1)

Right/Oblique triangle


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A trigonometric function is a ratio of certain parts of a triangle. The

names of these ratios are: The sine, cosine, tangent, cosecant, secant,

cotangent.

Let us look at this triangle

a c

b

A

B

C

Given the assigned letters to the sides and

angles, we can determine the following

trigonometric functions.

The Cosecant is the inversion of thesine, the secant is the inversion of

the cosine, the cotangent is the

inversion of the tangent.

With this, we can find the sine of the

value of angleA by dividing side aby side c. In order to find the angle

itself, we must take the sine of the

angle and invert it (in other words,

find the cosecant of the sine of the

angle).

Sin=

Cos =

Tan =

Side Opposite

SideAdjacent

SideAdjacentSide Opposite

Hypothenuse

Hypothenuse

=

=

= a

b

ca

b

c


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Try finding the angles of the following triangle from the

side lengths using the trigonometric ratios from the

previous slide.

610

8

A

B

C

The first step is to use the trigonometric

functions on angle A.

Sin =6/10

Sin =0.6

Csc0.6~36.9

Angle A~36.9

Because all angles add up to 180,

B=90-11.537=53.1

C

2

34A

B

The measurements have changed. Find side BA and sideAC

Sin34=2/BA

0.559=2/BA

0.559BA=2

BA=2/0.559

BA~3.578

The Pythagorean theorem

when used in this triangle states

that

BC2+AC2=AB2

AC2=AB2-BC2

AC2=12.802-4=8.802

AC=8.8020.5~3


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When solving oblique triangles, simply using

trigonometric functions is not enough. You need

The Law of Sines

C

c

B

b

A

a

sinsinsin

!!

The Law of Cosines

a2=b2+c2-2bc cosA

b2=a2+c2-2ac cosB

c2=a2+b2-2ab cosC

It is useful to memorize theselaws. They can be used to

solve any triangle if enough

measurements are given.

a

c

bA

B

C


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When solving a triangle, you must remember to choose

the correct law to solve it with.

Whenever possible, the law of sines should be used.

Remember that at least one angle measurement must be

given in order to use the law of sines.

The law of cosines in much more difficult and time

consuming method than the law of sines and is harder to

memorize. This law, however, is the only way to solve a

triangle in which all sides but no angles are given.

Only triangles with all sides, an angle and two sides, or a

side and two angles given can be solved.


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a=4

c=6

b

A

B

C

28

Solve this triangle

Because this triangle has an angle given, we can use the law of sines to solve it.

a/sinA = b/sin B = c/sin C and subsitute: 4/sin28 = b/sin B = 6/C. Because we know nothing about

b/sin B, lets start with 4/sin28 and use it to solve 6/sin C.

Cross-multiply those ratios: 4*sin C = 6*sin 28, divide 4: sin C = (6*sin28)/4.

6*sin28=2.817. Divide that by four: 0.704. This means that sin C=0.704. Find the Csc of 0.704 .

Csc0.704 =44.749. Angle C is about 44.749. Angle B is about 180-44.749-28=17.251.

The last side is b. a/sinA = b/sinB, 4/sin28 = b/sin17.251, 4*sin17.251=sin28*b,

(4*sin17.251)/sin28=b. b~2.53.


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a=2.4

c=5.2

b=3.5A

B

C

Solve this triangle:Hint: use the law of cosines

Start with the law of cosines because there are no angles given.

a2=b2+c2-2bc cosA. Substitute values. 2.42=3.52+5.22-2(3.5)(5.2) cosA,

5.76-12.25-27.04=-2(3.5)(5.2) cos A, 33.53=36.4cosA, 33.53/36.4=cos A, 0.921=cos A, A=67.07.

Now forB.

b2=a2+c2-2ac cosB, (3.5)2=(2.4)2+(5.2)2-2(2.4)(5.2) cosB, 12.25=5.76+27.04-24.96 cos B.

12.25=5.76+27.04-24.96 cos B, 12.25-5.76-27.04=-24.96 cos B. 20.54/24.96=cos B. 0.823=cos B.

B=34.61.

C=180-34.61-67.07=78.32.


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Trigonometric identities are ratios

and relationships between certain

trigonometric functions.

In the following few slides, you

will learn about different

trigonometric identities that take

place in each trigonometric

function.


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What is the sine of 60? 0.866. What is the cosine of30?

0.866. If you look at the name of cosine, you can actually

see that it is the cofunction of the sine (co-sine). The

cotangent is the cofunction of the tangent (co-tangent), andthe cosecant is the cofunction of the secant (co-secant).

Sine60=Cosine30

Secant60=Cosecant30

tangent30=cotangent60


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Sin =1/csc

Cos =1/sec

Tan =1/cot

Csc =1/sin

Sec =1/cos

Tan =1/cot

The following trigonometric identities are useful to remember.

(sin )2

+ (cos )2

=1

1+(tan )2=(sec )2

1+(cot )2=(csc )2


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Degrees and pi radians are two methods of

showing trigonometric info. To convert

between them, use the following equation.

2 radians = 360 degrees

1 radians= 180 degrees

Convert 500 degrees into radians.

2 radians =360 degrees, 1 degree = 1 radians/180,500 degrees = radians/180 * 500

500 degrees = 25 radians/9


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Write out the each of the trigonometric functions (sin, cos, and tan) of the followingdegrees to the hundredth place.

(In degrees mode). Note: you do not have to do all of them

1. 45

2. 38

3. 22

4. 18

5. 95

6. 63

7. 90

8. 152

9. 112

10. 58

11. 345

12. 221

13. 47

14. 442

15. 123

16. 53

17. 41

18. 22

19. 75

20. 34

21. 53

22. 92

23. 153

24. 1000


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Solve the following right triangles with the dimensions given

5

c

22A

B

C

9 20

18A

B

C

A

a

c

13

B

C

52

c

12

8 A

B

C


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Solve the following oblique triangles with the dimensions given

12

22

14A

B

C

a

25

b

28

A

B

C

31

15

c

24

35 A

B

C

5

c

8A

B

C

168


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1. 45

2. 38

3. 22

4. 18

5. 95

6. 63

7. 90

8. 152

9. 112

10. 58

11. 345

12. 221

13. 47

14. 442

15. 123

16. 53

17. 41

18. 22

19. 75

20. 34

21. 53

22. 92

23. 153

24. 1000

Find each sine, cosecant, secant, and cotangent using different

trigonometric identities to the hundredth place

(dont just use a few identities, try all of them.).


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Convert to radians

52

34

35

46

74

36

15

37

94

53

174

156

376

324

163

532

272

631

856

428

732

994

897

1768

2000


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Convert to degrees

3.2 rad

3.1 rad

1.3 rad

7.4 rad

6.7 rad

7.9 rad

5.4 rad

9.6 rad

3.14 rad

6.48 rad

8.23 rad

5.25 rad

72.45 rad

93.16 rad

25.73 rad

79.23 rad

52.652 rad

435.96 rad

14.995 rad

745.153 rad


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trigonometry! ;)

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