trigo basics

2.  the lengths of the sides (A, B, and C)
 the measures of the acute angles (a and
b)
 (The third angle is always 90 degrees)
b
C
A
a
B

3. A C2 B2
B C2 A2
C A2 B2
if A 3, B 4
C A2 B2 b
C
C 32 42 A=3
a
C 25 5 B=4

4.  This works because there are 180º in a
triangle and we are already using up 90º
 For example:
 if a = 30º
 b = 90º – 30º
b
 b = 60º C
A
a
B

5.  Well, here is the central insight of
trigonometry:
 If you multiply all the sides of a right
triangle by the same number (k), you get
a triangle that is a different size, but
which has the same angles:
k(C) b
C b A k(A)
a a
B
k(B)

6.  Take a triangle where angle b is 60º and
angle a is 30º
 If side B is 1unit long, then side C must be
2 units long, so that we know that for a
triangle of this shape the ratio of side B to
C is 1:2
 There are ratios for every
C=2 60 º
shape of triangle! A=1
30º
B

7.  Yes, so there are three sets of ratios for
any triangle
 They are mysteriously named:
 sin…short for sine
 cos…short for cosine
 tan…short or tangent
 and the ratios are already calculated,
you just need to use them

8. opp
sin Tan is Adjacent over Hypotenuse
Cos is Opposite over Adjacent
Sin is Opposite over Hypotenuse
hyp
adj SOHCAHTOA
cos
hyp
opp
tan
adj

9.  Before we can use the ratios we need to
get a few terms straight
 The hypotenuse (hyp) is the longest side
of the triangle – it never changes
 The opposite (opp) is the side directly
across from the angle you are
considering
 The adjacent (adj) is the side right beside
the angle you are considering

10.  looking at the triangle in terms of angle
b
 A is the adjacent b
C
(near the angle) A
 B is the opposite B
(across from the angle)
b Near
 C is always the Longest hyp
hypotenuse adj
opp Across

11.  looking at the triangle in terms of angle
a
 A is the opposite (across C
from the angle) A
a
 B is the adjacent (near B
the angle)
hyp Across
 C is always the Longest
hypotenuse opp
a
adj Near

12.  Suppose we want
to find angle a opp
 what is side A? tan
 the opposite adj
 what is side B?
 the adjacent
 with opposite and
adjacent we use C
b
the… A=3
 tan formula a
B=4

13. opp
tan
adj
3
tan a 0.75
4
check our calculator
s
b
a 36.87º C
A=3
a
B=4

14.  Each shape of triangle has three ratios
 These ratios are stored your scientific
calculator
 In the last question, tanθ = 0.75
 On your calculator try 2nd, Tan 0.75 =
36.87

15.  we want to find angle b
opp
 B is the opposite tan
 A is the adjacent adj
 so we use tan
4
tan b
3 b
C
tan b 1.33 A=3
a
b 53.13 B=4

16.  you know a side (adj) and an angle
(25 )
 we want to know the opposite side
opp
tan
adj
A
tan 25
6
A tan 25 6
b
A 0.47 6 C
A
A 2.80 25
B=6

17.  If you know a side and an angle, you
can find the other side.
6 opp
tan 25 tan
B adj
6
B
tan 25
b
6 C
B A=6
0.47 25
B 12.87 B

18.  You look up at an angle of 65° at the top of
a tree that is 10m away
 the distance to the tree is the adjacent side
 the height of the tree is the opposite side
opp
tan 65
10
opp 10 tan 65
65
opp 10 2.14
10m
opp 21.4

19.  We use sin and cos when we need to
work with the hypotenuse
 if you noticed, the tan formula does not
have the hypotenuse in it.
 so we need different formulas to do this
work
 sin and cos are the ones!10
C= b
A
25
B

20.  we want to find angle a
 since we have opp and hyp opp
we use sin
sin
hyp
5
sin a
10
C = 10 b
sin a 0 .5 A=5
a 30 a
B

21.  find the length of
side A opp
 We have the angle sin
hyp
and the hyp, and we
need the opp
A
sin 25
20
A sin 25 20
C = 20 b
A 0.42 20 A
A 8.45 25
B

22.  We use cos when we need to work with
the hyp and adj
adj
 so lets find angle b cos
hyp
4
cos b b
10 C = 10
A=4
cos b 0.4 a
b 66.42 B
a 90 - 66.42
a 23.58

23.  Spike wants to ride down a
steel beam
 The beam is 5m long and is
leaning against a tree at an
angle of 65 to the ground
 His friends want to find out
how high up in the air he is
when he starts so they can put
add it to the doctors report at
the hospital
 How high up is he?

24.  Well, what are we working
with?
 We have an angle
 We have hyp
C=5
 We need opp
B
 With these things we will use
the sin formula
65

25. opp
sin 65
hyp
opp
sin 65
5
C=5
opp sin 65 5
B
opp 0.91 5
opp 4.53
 so Spike will have fallen 65
4.53m

26.  Lucretia drops her
walkman off the Leaning
Tower of Pisa when she
visits Italy
 It falls to the ground 2
meters from the base of
the tower
 If the tower is at an angle
of 88 to the ground, how
far did it fall?

27.  What parts do we have?
 We have an angle
 We have the Adjacent
 We need the opposite
 Since we are working with B
the adj and opp, we will
use the tan formula
88
2m

28. opp
tan 88
adj
opp
tan 88
2
opp tan 88 2
opp 28.64 2 B
opp 57.27
 Lucretia’s walkman fell 57.27m 88
2m

29. 1. Make a diagram if needed
2. Determine which angle you are
working with
3. Label the sides you are working with
4. Decide which formula fits the sides
5. Substitute the values into the formula
6. Solve the equation for the unknown
value
7. Does the answer make sense?

30.  Although there are two triangles, you
only need to solve one at a time
 The big thing is to analyze the system to
understand what you are being given
 Consider the following problem:
 You are standing on the roof of one
building looking at another building, and
need to find the height of both buildings.

31.  You can measure
the angle 40°
down to the base
of other building
and up 60° to
the top as well. 60
You know the 40
distance
between the two
buildings is 45m 45m

32.  The first triangle:
a
60
 The second 45m
triangle 40
b
 notethat they
share a side 45m
long

33.  We are dealing with an angle, the
opposite and the adjacent
 this gives us Tan
a
tan 60
45
a tan 60 45
a
a 1.73 45
a 77.94m 60
45m

34.  We are dealing with an angle, the opposite
and the adjacent
 this gives us Tan
b
tan 40 45m
45
40
b tan 40 45
b
b 0.84 45
b 37.76m

35.  Look at the diagram
now:
 the short building is
37.76m tall 77.94m
 the tall building is
60
77.94m plus 37.76m
40
tall, which equals
115.70m tall 37.76m
45m