Trigonometry

1
Trigonometry
hypotenuse
Triangle terminology
opposite side
angle
adjacent side
Definitions
sine cosine tangent
a2 b2 c2
Pythagoras theorem
sin2(?) cos2(?) 1
Further topics
Exit
2
Triangle terminology
Opposite side
Adjacent side
Hypotenuse
Exit
Contents
D terminology
Definitions
3
Opposite side the side opposite the angle
angle
opposite
opposite
opposite
angle
angle
angle
opposite
Clear
Hypotenuse
Adjacent side
Exit
Contents
D terminology
Definitions
4
Adjacent side the side beside the angle
adjacent
angle
adjacent
angle
angle
adjacent
adjacent
angle
Opposite side
Clear
Hypotenuse
Exit
Contents
D terminology
Definitions
5
Hypotenuse the longest side
hypotenuse
hypotenuse
hypotenuse
hypotenuse
Opposite side
Clear
Adjacent side
Exit
Contents
D terminology
Definitions
6
Definitions
B
c
a
A
90o
C
b
Tangent
Sine
Cosine
Exit
Contents
D terminology
Definitions
7
Tangent of angle A
opposite adjacent
a b
B
tan(A)

c
a
tan(B)
A
90o
C
b
clear
Sine
Cosine
Exit
Contents
D terminology
Definitions
8
Tangent of angle B
opposite adjacent
b a
B
tan(B)

c
a
tan(A)
A
90o
C
b
clear
Sine
Cosine
Exit
Contents
D terminology
Definitions
9
Sine of angle A
a c
opposite hypotenuse
B
sin(A)

c
a
sin(B)
A
90o
C
b
Tangent
Cosine
Clear
Exit
Contents
D terminology
Definitions
10
Sine of angle B
b c
opposite hypotenuse
B
sin(B)

c
a
sin(A)
A
90o
C
b
Tangent
Clear
Cosine
Exit
Contents
D terminology
Definitions
11
Cosine of angle A
b c
adjacent hypotenuse
B
cos(A)

c
a
cos(B)
A
90o
C
b
Tangent
Sine
Clear
Exit
Contents
D terminology
Definitions
12
Cosine of angle B
a c
adjacent hypotenuse
B
cos(B)

c
a
cos(A)
A
90o
C
b
Tangent
Sine
Clear
Exit
Contents
D terminology
Definitions
13
Trigonometry further topics
Sine rule
Cosine rule
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
Identity
sin(AB) sin(A)cos(B) cos(A)sin(B)
Exit
Contents
Further topics
14
Sine Rule
A B
B C
B
c
a
A
C
b
Exit
Contents
Further topics
back
Clear
15
Sine Rule
B
sin(A)

c
a
C
b
Exit
Contents
Further topics
back
Clear
16
Sine Rule
B
h b
sin(A)

c
a
sin(A) a
gt
h

A
b
C
Exit
Contents
Further topics
back
Clear
17
Sine Rule
B
sin(B)
sin(A)


c
a
sin(A) a
h b a

gt
h
A
b
C
Exit
Contents
Further topics
back
Clear
18
Sine Rule
B
h a
sin(B)
sin(A)


c
a
sin(A) a
sin(B) b
h b a

gt
gt
h

A
b
C
Exit
Contents
Further topics
back
Clear
19
Sine Rule
B
sin(B)
sin(A)


a
c
sin(A) a
h b a
h a b

gt
gt

h
A
gt
b
C
Exit
Contents
Further topics
back
Clear
20
Sine Rule
B
sin(B)
sin(A)


a
c
sin(A) a
h b a
h a b

gt
gt

h
sin(B) b
sin(A) a
A

gt
b
C
Exit
Contents
Further topics
back
Clear
21
Sine Rule
B
sin(B)

c
a
C
b
Exit
Contents
Further topics
back
Clear
22
Sine Rule
B
h c
sin(B)

c
a
sin(B) b
gt

C
b
Exit
Contents
Further topics
back
Clear
23
Sine Rule
B
sin(C)
sin(B)


c
a
sin(B) b
h c b

gt
h
C
A
b
Exit
Contents
Further topics
back
Clear
24
Sine Rule
B
h b
sin(C)
sin(B)


c
a
sin(B) b
sin(C) c
h c b

gt
gt

h
C
A
b
Exit
Contents
Further topics
back
Clear
25
Sine Rule
B
sin(C)
sin(B)


c
a
sin(B) b
h c b
h b c

gt
gt

h
C
gt
A
b
Exit
Contents
Further topics
back
Clear
26
Sine Rule
B
sin(C)
sin(B)


c
a
sin(B) b
h c b
h b c

gt
gt

h
sin(C) c
sin(B) b
C

gt
A
b
Exit
Contents
Further topics
back
Clear
27
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
Show proof
P
Q
Exit
Contents
Further topics
back
Clear
28
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
Sine rule
gt sin(PQ)
p
q
P
Q
b
Exit
Contents
Further topics
back
Clear
29
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
sin(B) b
gt sin(PQ)
(pq)
p
c
gt sin(PQ)
Substitute sin(B) cos(P)
h
q
P
Q
b
Exit
Contents
Further topics
back
Clear
30
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
sin(B) b
gt sin(PQ)
(pq)
p
c
cos(P) b
gt sin(PQ)
(pq)
h
q
P
gt sin(PQ)
Expand bracket
Q
b
Exit
Contents
Further topics
back
Clear
31
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
sin(B) b
gt sin(PQ)
(pq)
p
c
cos(P) b
gt sin(PQ)
(pq)
h
q
P
gt sin(PQ)
Q
b
b
Substitute cos(P) h/c q/b sin(Q)
gt sin(PQ)
Exit
Contents
Further topics
back
Clear
32
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
sin(B) b
gt sin(PQ)
(pq)
p
c
cos(P) b
gt sin(PQ)
(pq)
h
q
P
p b
q b
gt sin(PQ)
cos(P)
cos(P)

Q
b
p b
h c
gt sin(PQ)
sin(Q)

x
cos(P)
Substitute p/c sin(P) h/b cos(Q)
gt sin(PQ)
Exit
Contents
Further topics
back
Clear
33
sin(PQ)
sin(P) cos(Q) sin(Q) cos(P)
B
sin(B) b
gt sin(PQ)
(pq)
p
c
cos(P) b
gt sin(PQ)
(pq)
h
q
P
p b
q b
gt sin(PQ)
cos(P)
cos(P)

Q
b
p b
h c
gt sin(PQ)
sin(Q)

x
cos(P)
gt sin(PQ)
sin(Q)cos(P)

sin(P)cos(Q)
Exit
Contents
Further topics
back
Clear
34
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
sin(A)
c

a
A
C
b
Exit
Contents
Further topics
back
Clear
35
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
a c
sin(A)
c


a
x
A
A
C
b
Exit
Contents
Further topics
back
Clear
36
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
a c
y a
sin(A)
c

gt
a2 cy

a
x
A
cos(A)
A

C
b
Exit
Contents
Further topics
back
Clear
37
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
sin(A)
c

gt
a2 cy

a
x
b c
cos(A)

A

C
b
Exit
Contents
Further topics
back
Clear
38
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
sin(A)
c

gt
a2 cy

a
x
b c
x b
cos(A)

gt
b2 cx

A
C
b
gt
Exit
Contents
Further topics
back
Clear
39
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
sin(A)
c

gt
a2 cy

a
x
cos(A)

gt
b2 cx

A
C
b
gt
a2 b2 cy cx
Note yxc gt c(y x ) c2
gt
Exit
Contents
Further topics
back
Clear
40
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
a c
y a
sin(A)
c

gt
a2 cy

a
x
b c
x b
cos(A)

gt
b2 cx

A
C
b
gt
a2 b2 cy cx
Note yxc gt c(y x ) c2
a2 b2 c2
gt
gt
Exit
Contents
Further topics
back
Clear
41
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
a c
y a
sin(A)
c

gt
a2 cy

a
x
b c
x b
cos(A)

gt
b2 cx

A
C
b
gt
a2 b2 cy cx c(y x ) c2
gt
a2 b2 c2
a2
b2
c2


gt
gt
Exit
Contents
Further topics
back
Clear
42
Pythagoras theorem
a2 b2 c2
sin2(A) cos2(A) 1
B
y
a c
y a
sin(A)
c

gt
a2 cy

a
x
b c
x b
cos(A)

gt
b2 cx

A
C
b
gt
a2 b2 cy cx c(y x ) c2
gt
a2 b2 c2
a2
b2
c2


gt
c2
c2
c2
cos2(A)
sin2(A)

1
gt
Exit
Contents
Further topics
back
Clear
43
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
Show proof
c
a
A
C
b
Exit
Contents
Further topics
back
Clear
44
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
Pythagoras Theorem
a2

c
a
h
A
x
b-x
C
b
Exit
Contents
Further topics
back
Clear
45
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
Expand bracket
a2
gt

c
a
h
A
x
b-x
C
b
Exit
Contents
Further topics
back
Clear
46
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
a2


h2
gt
b2 - 2bx
x2
c
a
Substitute h2 c2 - x2
h
a2
gt

A
x
b-x
C
b
Note h2 c2 - x2
Exit
Contents
Further topics
back
Clear
47
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
a2


h2
gt
b2 - 2bx
x2
c
a
h
a2

b2 - 2bx

c2
gt
- x2
x2
Simplify
a2
gt

A
x
b-x
C
b
Note h2 c2 - x2
Exit
Contents
Further topics
back
Clear
48
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
a2


h2
gt
b2 - 2bx
x2
c
a
h
a2

b2 - 2bx

c2
gt
- x2
x2
a2

b2 - 2bx

c2
gt
A
x
b-x
C
Substitute x c cos(A)
b
a2
gt

Note x c cos(A)
Exit
Contents
Further topics
back
Clear
49
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
a2


h2
gt
b2 - 2bx
x2
c
a
h
a2

b2 - 2bx

c2
gt
- x2
x2
a2

b2 - 2bx

c2
gt
A
x
b-x
C
b
a2

b2 - 2bc cos(A)

c2
gt
Note x c cos(A)
a2

gt
Rearrange
Exit
Contents
Further topics
back
Clear
50
Cosine Rule
a2 b2 c2 - 2bc cos(A)
B
a2

(b-x)2

h2
a2


h2
gt
b2 - 2bx
x2
c
a
h
a2

b2 - 2bx

c2
gt
- x2
x2
a2

b2 - 2bx

c2
gt
A
x
b-x
C
b
a2

b2 - 2bc cos(A)

c2
gt
a2

b2
c2
gt
- 2bc cos(A)
Exit
Contents
Further topics
back
Clear
51
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
a
A
C
b
Next
Exit
Contents
Further topics
back
Clear
52
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
height
a
A
C
b
base
Area of rectangle
base x height

Area of triangle ABC

ready
Next
Exit
Contents
Further topics
back
Clear
53
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
height
a
A
C
b
base
Area of rectangle
base x height

Area of triangle ABC
½ of LEFT rect. ½ of RIGHT rect.

Area of triangle
½ base x height

ready
Next
Exit
Contents
Further topics
back
Clear
54
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
height
a
A
C
b
base
Area of triangle
½ base x height

ready
Next
Exit
Contents
Further topics
back
Clear
55
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
height
a
A
C
b
base
Area of triangle
½ base x height

height
Area of triangle
½ base x

c
x
c
ready
Next
Exit
Contents
Further topics
back
Clear
56
Area of triangle
Area ½ a b sin(C) ½ b c sin(A) ½ c a sin(B)
B
c
height
a
A
C
b
base
Area of triangle
½ base x height

height
Area of triangle
½ base x

c
x
c
Area of triangle
½

c
Sin(A)
b
complete
Exit
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Further topics
back
Clear