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Trigonometry
(4103)
Trigonometry
“triangle measure”
A little bit of review...
a
The 3 angles from
a triangle ALWAYS
equal 180o
b
a + b + c = 180o
c
Find the total of the other angles
a
30◦
Find the total of the other angles
a
= 90◦
30◦
Find the total of the other angles
Total angles = 180◦
90◦ + 30◦ + a = 180◦
120◦ + a = 180◦
a = 180◦ – 120◦
a = 60◦
a
= 90◦
30◦
Equilateral triangle
All sides are the
same length
Equilateral triangle
All angles
are the same
(180o ÷ 3 = 60o)
Isosceles triangle
Two sides are
the same length
Isosceles triangle
Two angles
are the same
Scalene triangle
No sides are the
same length
Scalene triangle
No angles are
the same
Right-angled triangle
side
hypotenuse
side
Right-angled triangle
side
opposite
to angle A
hypotenuse
A
side adjacent
(next to) angle A
Right-angled triangle
hypotenuse (c)
side (a)
90o
side (b)
Pythagorean Theorem
c 2 = a 2 + b2
side (a)
hypotenuse (c)
side (b)
What if you switch a and b?
c 2 = a 2 + b2
side (a)
hypotenuse (c)
side (b)
What if you switch a and b?
c 2 = a 2 + b2
side (b)
hypotenuse (c)
side (a)
Doesn’t
matter,
they’re
both sides!
Right-angled triangle
B
side
adjacent
to angle B
hypotenuse
A
side opposite
to angle B
What is the length of the hypotenuse?
c 2 = a 2 + b2
side (a)
3 cm
hypotenuse (c)
x cm
side (b)
4 cm
What is the length of the hypotenuse?
c2 = a 2 + b 2
side (a)
3 cm
hypotenuse (c)
x cm
side (b)
4 cm
x2 = 32 + 42
x2 = 9 + 16
x2 = 25
x2 = 25
x = 5 cm
What is the length of the side?
c 2 = a 2 + b2
side (a)
x cm
hypotenuse (c)
10 cm
side (b)
5 cm
What is the length of the side?
c2 = a 2 + b 2
side (a)
x cm
hypotenuse (c)
10 cm
side (b)
5 cm
102 = x2 + 52
100 = x2 + 25
100 – 25 = x2
75 = x2
x2 = 75
x = 8.7 cm
Trigonometric ratios
sine
cosine
tangent
depend on
which angle is used
Trigonometric ratios
sine
cosine
tangent
depend on
which angle is used
Sine ratio (SOH)
sin A = opposite
hypotenuse
side
opposite
to angle A
hypotenuse
A
side adjacent
to angle A
Sine ratio (SOH)
sin B = opposite
hypotenuse
B
side
adjacent
to angle B
hypotenuse
A
side opposite
to angle B
Cosine ratio (CAH)
cos A = adjacent
hypotenuse
side
opposite
to angle A
hypotenuse
A
side adjacent
to angle A
Cosine ratio (CAH)
cos B = adjacent
hypotenuse
B
side
adjacent
to angle B
hypotenuse
A
side opposite
to angle B
Tangent ratio (TOA)
tan A = opposite
adjacent
side
opposite
to angle A
hypotenuse
A
side adjacent
to angle A
Tangent ratio (TOA)
tan B = opposite
adjacent
B
side
adjacent
to angle B
hypotenuse
A
side opposite
to angle B
Trigonometric ratios
SOH CAH TOA
sin θ = opp cos θ = adj tan θ = opp
hyp
adj
hyp
Find the lengths of the missing sides
and angle (right triangle)
B
7 cm
35o
A
C
Find the lengths of the missing sides
and angle (right triangle)
B
7 cm
35o
A
90o
C
Step 1. List the information given, and
what is needed
B
What we know:
mBC = 7 cm
 A = 35o
 C = 90o
7 cm
35o
A
90o
C
What we need:
mAB = ?
mAC = ?
B = ?
Step 2. Find the missing side AB
B
hyp
7 cm
(opp)
35o
A
90o
C
Look at the
triangle from  A:
mBC = opposite
mAB = hypotenuse
Step 2. Find the missing side AB
B
hyp
Look at the
triangle from  A:
mBC = opposite
mAB = hypotenuse
7 cm
(opp)
35o
A
90o
C
? = opp
hyp
Step 2. Find the missing side AB
B
hyp
Look at the
triangle from  A:
mBC = opposite
mAB = hypotenuse
7 cm
(opp)
35o
A
90o
C
sin θ = opp
hyp
Step 2. Find the missing side AB
B
hyp
7 cm
(opp)
35o
A
90o
C
sin θ = opp
hyp
sin 35o = opp
hyp
Step 2. Find the missing side AB
B
hyp
7 cm
(opp)
35o
A
90o
C
sin θ = opp
hyp
sin 35o = opp
hyp
0.574 = 7 cm
hyp
Step 2. Find the missing side AB
B
hyp
7 cm
(opp)
35o
A
90o
C
sin θ = opp
hyp
sin 35o = opp
hyp
0.574 = 7 cm
hyp
Step 2. Find the missing side AB
sin θ = opp
hyp
B
hyp
7 cm
(opp)
35o
A
90o
C
sin 35o = opp
hyp
0.574 = 7 cm
hyp
0.574 (hyp) = 7 cm
Step 2. Find the missing side AB
sin θ = opp
hyp
B
hyp
7 cm
(opp)
35o
A
90o
C
sin 35o = opp
hyp
0.574 = 7 cm
hyp
0.574 (hyp) = 7 cm
0.574
0.574
Step 2. Find the missing side AB
sin θ = opp
hyp
B
hyp
7 cm
(opp)
35o
A
90o
C
sin 35o = opp
hyp
0.574 = 7 cm
hyp
0.574 (hyp) = 7 cm
hyp = 12.2 cm
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
Look at the
triangle from  A:
mBC = opposite
mAB = hypotenuse
mAC = adjacent
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
Look at the
triangle from  A:
mBC = opposite
mAB = hypotenuse
mAC = adjacent
Since we have two
sides, we have a
choice of trig ratios!
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos θ = adj
hyp
or
tan θ = opp
adj
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos θ = adj
hyp
cos 35o = adj
hyp
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos θ = adj
hyp
cos 35o = adj
hyp
0.819 = adj
12.2 cm
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos θ = adj
hyp
cos 35o = adj
hyp
0.819 = adj
12.2 cm
Step 3. Find the missing side AC
cos θ = adj
hyp
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos 35o = adj
hyp
0.819 = adj
12.2 cm
0.819 (12.2 cm) = adj
Step 3. Find the missing side AC
cos θ = adj
hyp
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
cos 35o = adj
hyp
0.819 = adj
12.2 cm
0.819 (12.2 cm) = adj
adj = 10 cm
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
tan θ = opp
adj
tan 35o = opp
adj
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
tan θ = opp
adj
tan 35o = opp
adj
0.700 = 7 cm
adj
Step 3. Find the missing side AC
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
tan θ = opp
adj
tan 35o = opp
adj
0.700 = 7 cm
adj
Step 3. Find the missing side AC
tan θ = opp
adj
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
tan 35o = opp
adj
0.700 = 7 cm
adj
0.700 (adj) = 7 cm
Step 3. Find the missing side AC
tan θ = opp
adj
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
(adj)
C
tan 35o = opp
adj
0.700 = 7 cm
adj
0.700 (adj) = 7 cm
0.700
0.700
Step 3. Find the missing side AC
tan θ = opp
adj
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
tan 35o = opp
adj
0.700 = 7 cm
adj
0.700 (adj) = 7 cm
adj = 10 cm
Step 4. Find the missing angle B
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
Step 4. Find the missing angle B
180o =  A +  B +  C
B
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
Step 4. Find the missing angle B
180o =  A +  B +  C
B
180o = 35o +  B + 90o
12.2 cm
(hyp)
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
Step 4. Find the missing angle B
180o =  A +  B +  C
B
180o = 35o +  B + 90o
12.2 cm
(hyp)
o =  B + 125o
180
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
Step 4. Find the missing angle B
180o =  A +  B +  C
B
180o = 35o +  B + 90o
12.2 cm
(hyp)
o =  B + 125o
180
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
 B = 180o – 125o
Step 4. Find the missing angle B
180o =  A +  B +  C
B
12.2 cm
(hyp)
180o = 35o +  B + 90o
55o
o =  B + 125o
180
7 cm
(opp)
35o
A
90o
10 cm
(adj)
C
 B = 180o – 125o
 B = 55o
Steps to completing a right triangle
Step 1. List the information given, and
what is needed
Step 2. Find the missing side(s)
Step 3. Find the missing angle(s)
Find the length of the missing side and
angles
A
25 cm
19 cm
30o
B
C
Step 1. List the missing information, and
what is needed
A
25 cm
19 cm
30o
B
C
What we know:
mAB = 25 cm
mAC = 19 cm
 B = 30o
What we need:
mBC = ?
 A=?
 C=?
Step 2. Create a 90o angle by cutting the
triangle in two
A
25 cm
19 cm
30o
B
H
C
Start at the top
angle and
continue until it
hits the bottom of
the triangle at a
90o angle
Step 2. Create a 90o angle by cutting the
triangle in two
Name the point of
intersection H
A
25 cm
19 cm
30o
B
H
C
Step 2. Create a 90o angle by cutting the
triangle in two
Name the point of
intersection H
A
25 cm
19 cm
30o
B
H
C
Now find the missing
information for each
new triangle!
Step 3. Find the length BH
Look at the new
triangle from  B:
mBH = adjacent
mAB = hypotenuse
A
25 cm
19 cm
30o
B
H
C
Step 3. Find the length BH
Look at the new
triangle from  B:
mBH = adjacent
mAB = hypotenuse
A
25 cm
19 cm
? = adj
hyp
30o
B
H
C
Step 3. Find the length BH
Look at the new
triangle from  B:
mBH = adjacent
mAB = hypotenuse
A
25 cm
19 cm
cos θ = adj
hyp
30o
B
H
C
Step 3. Find the length BH
cos θ = adj
hyp
A
25 cm
19 cm
30o
B
H
C
cos 30o = adj
hyp
Step 3. Find the length BH
cos θ = adj
hyp
A
25 cm
19 cm
30o
B
H
C
cos 30o = adj
hyp
0.866 = adj
25 cm
Step 3. Find the length BH
cos θ = adj
hyp
A
25 cm
19 cm
30o
B
H
C
cos 30o = adj
hyp
0.866 = adj
25 cm
Step 3. Find the length BH
cos θ = adj
hyp
A
25 cm
30o
B
H
cos 30o = adj
19 cm
hyp
0.866 = adj
25 cm
(0.866)(25 cm) = adj
C
Step 3. Find the length BH
cos θ = adj
hyp
A
25 cm
30o
B
21.7 cm
H
cos 30o = adj
19 cm
hyp
0.866 = adj
25 cm
(0.866)(25 cm) = adj
C
adj = 21.7 cm
Step 4. Find the angle A
A
25 cm
180o =  A +  B +  H
19 cm
30o
B
21.7 cm
H
C
Step 4. Find the angle A
A
180o =  A +  B +  H
180o =  A + 30o + 90o
25 cm
19 cm
30o
B
21.7 cm
H
C
Step 4. Find the angle A
A
180o =  A +  B +  H
180o =  A + 30o + 90o
25 cm
19 cm
30o
B
21.7 cm
H
C
180o =  A + 120o
Step 4. Find the angle A
A
180o =  A +  B +  H
180o =  A + 30o + 90o
25 cm
19 cm
30o
B
21.7 cm
H
C
180o =  A + 120o
 A = 180o – 120o
Step 4. Find the angle A
A
180o =  A +  B +  H
180o =  A + 30o + 90o
60o
25 cm
19 cm
21.7 cm
 A = 180o – 120o
 A = 60o
30o
B
180o =  A + 120o
H
C
Step 5. Find the length AH
A
60o
25 cm
19 cm
30o
B
21.7 cm
H
C
There are many
different ways to
find mAH:
– Pythagoras
– tan A or tan B
– cos A
– sin B
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
19 cm
30o
B
21.7 cm
H
C
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
19 cm
30o
B
21.7 cm
H
C
sin 30o = opp
hyp
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
19 cm
30o
B
21.7 cm
H
C
sin 30o = opp
hyp
0.500 = opp
25 cm
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
19 cm
30o
B
21.7 cm
H
C
sin 30o = opp
hyp
0.500 = opp
25 cm
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
30o
B
21.7 cm
H
sin 30o = opp
19 cm
hyp
0.500 = opp
25 cm
(0.500)(25) = opp
C
Step 5. Find the length AH
sin θ = opp
hyp
A
60o
25 cm
12.5
cm
30o
B
21.7 cm
H
sin 30o = opp
19 cm
hyp
0.500 = opp
25 cm
(0.500)(25) = opp
C
hyp = 12.5 cm
Step 6. Find the angle C
Look at the new
triangle from  C:
mAH = opposite
mAC = hypotenuse
A
60o
25 cm
19 cm
12.5
cm
? = opp
hyp
30o
B
21.7 cm
H
C
Step 6. Find the angle C
Look at the new
triangle from  C:
mAH = opposite
mAC = hypotenuse
A
60o
25 cm
19 cm
12.5
cm
sin θ = opp
hyp
30o
B
21.7 cm
H
C
Step 6. Find the angle C
sin θ = opp
hyp
A
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
H
C
Step 6. Find the angle C
sin θ = opp
hyp
A
sin θ = opp
hyp
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
H
C
Step 6. Find the angle C
sin θ = opp
hyp
A
sin θ = opp
hyp
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
H
C
sin θ = 12.5 cm
19 cm
Step 6. Find the angle C
sin θ = opp
hyp
A
sin θ = opp
hyp
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
H
C
sin θ = 12.5 cm
19 cm
sin θ = 0.66
Step 6. Find the angle C
sin θ = opp
hyp
A
sin θ = opp
hyp
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
H
C
sin θ = 12.5 cm
19 cm
sin θ = 0.66
sin-1(0.66) = θ
Step 6. Find the angle C
sin θ = opp
hyp
A
sin θ = opp
hyp
60o
25 cm
19 cm
12.5
cm
41.1o
30o
B
21.7 cm
H
C
sin θ = 12.5 cm
19 cm
sin θ = 0.66
sin-1(0.66) = θ
θ = 41.1o
Step 7. Find the length CH
A
60o
25 cm
19 cm
12.5
cm
41.1o
30o
B
21.7 cm
H
C
There are many
different ways to
find mCH:
– Pythagoras
– cos C
– tan C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
19 cm
12.5
cm
41.1o
30o
B
21.7 cm
H
C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
cos 41.1o = adj
hyp
19 cm
12.5
cm
41.1o
30o
B
21.7 cm
H
C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
12.5
cm
41.1o
30o
B
21.7 cm
cos 41.1o = adj
hyp
19 cm
0.754 = adj
19 cm
H
C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
12.5
cm
41.1o
30o
B
21.7 cm
cos 41.1o = adj
hyp
19 cm
0.754 = adj
19 cm
H
C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
12.5
cm
30o
B
21.7 cm
H
cos 41.1o = adj
hyp
19 cm
0.754 = adj
19 cm
41.1o
0.754 (19 cm) = adj
C
Step 7. Find the length CH
cos θ = adj
hyp
A
60o
25 cm
12.5
cm
30o
B
21.7 cm
cos 41.1o = adj
hyp
19 cm
0.754 = adj
19 cm
41.1o
0.754 (19 cm) = adj
H 14.3 cm C
adj = 14.3 cm
Step 8. Find the angle A
180o =  A +  C +  H
A
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
Step 8. Find the angle A
180o =  A +  C +  H
A
180o =  A + 41.1o + 90o
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
Step 8. Find the angle A
180o =  A +  C +  H
A
180o =  A + 41.1o + 90o
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
180o =  A + 131.1o
Step 8. Find the angle A
180o =  A +  C +  H
A
180o =  A + 41.1o + 90o
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
180o =  A + 131.1o
 A = 180o – 131.1o
Step 8. Find the angle A
180o =  A +  C +  H
A
48.9o
180o =  A + 41.1o + 90o
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
180o =  A + 131.1o
 A = 180o – 131.1o
 A = 48.9o
Step 9. Complete triangle
A = 60o + 48.9o
= 108.9o
180o =  A +  B +  C
A
48.9o
60o
25 cm
19 cm
12.5
cm
30o
B
21.7 cm
41.1o
H 14.3 cm C
Step 9. Complete triangle
A = 60o + 48.9o
= 108.9o
25 cm
30o
B
21.7 cm
180o =  A +  B +  C
A
108.9o
180o = 108.9o + 30o + 41.1o
19 cm
41.1o
H 14.3 cm C
Step 9. Complete triangle
A = 60o + 48.9o
= 108.9o
25 cm
30o
B
21.7 cm
180o =  A +  B +  C
A
108.9o
180o = 108.9o + 30o + 41.1o
19 cm
41.1o
H 14.3 cm C
The angles in the
original triangle
ABC add up
to 180o
Step 9. Complete triangle
mBC = mBH + mCH
A
108.9o
25 cm
30o
B
21.7 cm
19 cm
41.1o
H 14.3 cm C
Step 9. Complete triangle
mBC = mBH + mCH
A
108.9o
25 cm
30o
B
21.7 cm
19 cm
41.1o
H 14.3 cm C
mBC = 21.7 + 14.3
Step 9. Complete triangle
mBC = mBH + mCH
A
108.9o
25 cm
19 cm
41.1o
30o
B
mBC = 21.7 + 14.3
36 cm
C
mBC = 36 cm
Steps to complete a non-right angle triangle
Step 1. List the missing information, and what is
needed
Step 2. Create 90o angles by cutting the triangle
in two
Step 3. Looking at the first triangle, solve for
missing angle(s) and/or side(s)
Step 4. Looking at the second triangle, solve for
missing angle(s) and/or side(s)
Step 5. Put the halves of sides and angles
together into the one original triangle
So far, there are two ways
to solve a right-angled
triangle:
So far, there are two ways
to solve a right-angled
triangle:
Pythagoras (c2 = a2 + b2)
Trigonometric ratios (SOH CAH TOA)
Isn’t there another way to
solve a non-right angled
triangle?
Isn’t there another way to
solve a non-right angled
triangle?
Yes!
Sin Law and Cos Law
Sine Law
Uses the sine ratio
Sine Law
a = b = c
sin A sin B sin C
Sine Law
lengths
a = b = c
sin A sin B sin C
angles
Find the length of the missing side and
angles
A
25 cm
19 cm
30o
B
C
Find the length of the missing side and
angles
Remember:
Capital letters = angles
Lower-case letters = sides
A
c
25 cm
b
19 cm
30o
B
a
C
Find the length of the missing side and
angles
angle A ↔ side a
Remember:
Capital letters = angles
Lower-case letters = sides
A
c
25 cm
b
19 cm
Angles and sides with the
same letters are opposite
each other
30o
B
a
C
Find the length of the missing side and
angles
angle B ↔ side b
Remember:
Capital letters = angles
Lower-case letters = sides
A
c
25 cm
b
19 cm
Angles and sides with the
same letters are opposite
each other
30o
B
a
C
Find the length of the missing side and
angles
angle C ↔ side c
Remember:
Capital letters = angles
Lower-case letters = sides
A
c
25 cm
b
19 cm
Angles and sides with the
same letters are opposite
each other
30o
B
a
C
Step 1. List the missing information,
and what is needed
A
c
25 cm
b
19 cm
30o
B
a
C
What we know:
mAB = c = 25 cm
mAC = b = 19 cm
 B = 30o
What we need:
mBC = a = ?
 A=?
 C=?
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
We have both
angle B and side b
A
c
25 cm
b
19 cm
30o
B
a
C
We can use these to
fill out the C ‘pair’
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
b = c
sin B sin C
A
b
19 cm
30o
B
a
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
A
b
19 cm
30o
B
a
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
A
b
19 cm
30o
B
a
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
A
b
19 cm
30o
B
a
C
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
19 (sin C) = sin 30o (25)
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
A
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
30o
B
a
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
A
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
19 (sin C) = 12.5
30o
B
a
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
30o
B
a
A
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
19 (sin C) = 12.5
19
19
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
30o
B
a
A
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
19 (sin C) = 12.5
19
19
sin C = 0.658
C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
30o
B
a
A
b = c
sin B sin C
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
19 (sin C) = 12.5
19
19
sin C = 0.658
C
sin-1 (0.658) = C
Step 2. Find one ‘pair’, and use it to fill
in another ‘pair’
a = b = c
sin A sin B sin C
c
25 cm
30o
B
a
b = c
sin B sin C
A
19 cm = 25 cm
sin 30o sin C
b
19 (sin C) = sin 30o (25)
19 cm
19 (sin C) = (0.5)(25)
19 (sin C) = 12.5
19
19
o
41.1
sin C = 0.658
C
sin-1 (0.658) = C
 C = 41.1o
Step 3. Find the last angle (A)
180o =  A +  B +  C
A
c
25 cm
b
19 cm
41.1o
30o
B
a
C
Step 3. Find the last angle (A)
180o =  A +  B +  C
A
180o =  A + 30o + 41.1o
c
25 cm
b
19 cm
41.1o
30o
B
a
C
Step 3. Find the last angle (A)
180o =  A +  B +  C
A
180o =  A + 30o + 41.1o
c
25 cm
b
19 cm
41.1o
30o
B
a
C
180o =  A + 71.1o
Step 3. Find the last angle (A)
180o =  A +  B +  C
A
180o =  A + 30o + 41.1o
c
25 cm
b
19 cm
180o =  A + 71.1o
180o – 71.1o =  A
41.1o
30o
B
a
C
Step 3. Find the last angle (A)
180o =  A +  B +  C
A
108.9o
c
25 cm
180o =  A + 30o + 41.1o
b
19 cm
180o =  A + 71.1o
180o – 71.1o =  A
B
A = 108.9o
41.1o
30o
a
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
b
19 cm
41.1o
30o
B
a
C
Step 4. Find the last ‘pair’ (A)
a = b
sin A sin B
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
b
19 cm
41.1o
30o
B
a
C
Step 4. Find the last ‘pair’ (A)
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
b
19 cm
41.1o
30o
B
a
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
41.1o
30o
B
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
41.1o
30o
B
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
41.1o
30o
B
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a (0.5) = 0.946 (19)
a
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
30o
B
a
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a (0.5) = 0.946 (19)
a (0.5) = 17.97
41.1o
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
30o
B
a
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a (0.5) = 0.946 (19)
a (0.5) = 17.97
41.1o
0.5
0.5
C
Step 4. Find the last ‘pair’ (A)
a = b = c
sin A sin B sin C A
108.9o
c
25 cm
30o
B
a
36 cm
a = b
sin A sin B
a
= 19 cm
sin 108.9o sin 30o
b
a = 19 cm
19 cm
0.946
0.5
a (0.5) = 0.946 (19)
a (0.5) = 17.97
41.1o
0.5
0.5
C
a = 36 cm
Done!
A
108.9o
c
25 cm
b
19 cm
41.1o
30o
B
a
36 cm
C
Steps to complete a triangle using
Sine Law
Step 1. List the missing information, and
what is needed
Step 2. Find one ‘pair’, and use it to fill in
another ‘pair’
Step 3. Find the last angle
Step 4. Find the last ‘pair’
Cos Law
Uses the cos ratio
Also uses ‘pairs’
Looking for a
Cos Law
other two lengths
2
a
=
2
b
+
2
c
– 2bc(cosA)
pair you’re looking for
Looking for b
Cos Law
other two lengths
2
b
=
2
a
+
2
c
– 2ac(cosB)
pair you’re looking for
Looking for c
Cos Law
other two lengths
2
c
=
2
b
+
2
a
– 2ab(cosC)
pair you’re looking for
3 variations of
Cos Law
2
a
2
b
2
c
= + – 2bc(cosA)
2
2
2
b = a + c – 2ac(cosB)
2
2
2
c = b + c – 2ab(cosC)
Find the length of b
A
c
25 cm
b
30o
B
a
36 cm
C
Find the length of b
A
c
25 cm
To use Cos Law, you have
to know:
- One of the values of the
pair you need
(angle or length)
- The two other lengths
b
30o
B
a
36 cm
C
Step 1. List the missing information,
and what is needed
What we know:
mAB = c = 25 cm
mBC = a = 36 cm
 B = 30o
A
c
25 cm
b
What we need:
mAC = b = ?
30o
B
a
36 cm
C
Step 2. Choose the variation of
Cos Law that you need
a2 = b2 + c2 – 2bc(cosA)
A
b2 = a2 + c2 – 2ac(cosB)
c
25 cm
c2 = b2 + c2 – 2ab(cosC)
b
30o
B
a
36 cm
C
Step 2. Choose the variation of
Cos Law that you need
a2 = b2 + c2 – 2bc(cosA)
A
b2 = a2 + c2 – 2ac(cosB)
c
25 cm
c2 = b2 + c2 – 2ab(cosC)
b
30o
B
a
36 cm
C
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 2(36)(25)(0.866)
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 1558.8
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 1558.8
b2 = 362.2
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 1558.8
b2 = 362.2
Step 3. Solve the equation
b2 = a2 + c2 – 2ac(cosB)
b2 = 362 + 252 – 2(36)(25)(cos30o)
b2 = 362 + 252 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 2(36)(25)(0.866)
b2 = 1296 + 625 – 1558.8
b2 = 362.2
b = 19 cm
Done!
A
c
25 cm
b
19 cm
30o
B
a
36 cm
C
Steps to complete triangles
using Cos Law
Step 1. List the missing information, and what is
needed
Step 2. Choose the variation of Cos Law that you
need
Step 3. Solve the equation
How do you know which to use?
Use Sin Law if:
Use Cos Law if:
• Given 2 sides, 1 angle
opposite one of the sides
• Given 3 sides
a
a
A
b
• Given 2 angles, 1 side
opposite one of the angles
B
b
• Given 1 angle, 2 sides
adjacent to that angle
a
c
C
c
C
b
Summary of trigonometry
Right-angled triangle
• If you only have sides
--Pythagoras
Non-right angled triangle
• If you have a pair (a + A)
• If you have sides and angles
• If you have to fill in a pair
(looking for angle or side)
– SOH CAH TOA
– Sin Law
– Cos Law
– a2 = b2 + c2 – 2ac(cos A)