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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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8/8/2019 Trigonometry! ;)
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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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8/8/2019 Trigonometry! ;)
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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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8/8/2019 Trigonometry! ;)
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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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8/8/2019 Trigonometry! ;)
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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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8/8/2019 Trigonometry! ;)
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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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