Transcript Trigonometry - Suffolk Maths

Trigonometry
Right angled triangles
A triangle
Opposite and Adjacent
are relative to the angle
• The 4cm side
is opposite to
A
• The 6cm side
is adjacent
to A
• The 6cm side
is opposite to
B
• The 4cm side
is adjacent
to B
SOH CAH TOA
Sine, Cosine & Tangent of
an angle
SOH CAH TOA
• SOH
– Some Old Houses
• (Sine of angle =Opposite side / Hypotenuse)
• CAH
– Creak And Howl
• (Cosine of angle = Adjacent side / hypotenuse)
• TOA
– Through Out Autumn
• (Tangent of angle = Opposite side / adjacent side)
TOA SOH CAH
• TOA
– Tom’s Old Auntie
• Tangent of angle = Opposite side / adjacent side
• SOH
– Sat On Him
• Sine of angle =Opposite side / Hypotenuse
• CAH
– Cursing At Him
• Cosine of angle = Adjacent side / hypotenuse
SOH CAH TOA
• SOH
SOH
Opposite side = sin (x)
times hypotenuse
Hypotenuse = opposite
side divided by sin (x)
Sin (x) = opposite side
divided by hypotenuse
SOH CAH TOA
CA H
• CAH
Adjacent side = cos (x)
times hypotenuse
Hypotenuse = adjacent
side divided by cos (x)
Cos (x) = adjacent side
divided by hypotenuse
SOH CAH TOA
• TOA
TO A
Opposite side = tan (x)
times adjacent side
Adjacent side = opposite
side divided by tan (x)
tan (x) = opposite side
divided by adjacent side
Problems for trig
A stage, is to be built for a concert, it has to be 2m high so
the audience can see the show.
The equipment needs to be pushed up onto the stage. Health
and safety rules say that a ramp must have a slope of no
more than 15 degrees.
The crew need to work out how far away from the stage to
start building the ramp.
m
A stage, is to be built for a concert, has to be 2m high so the
audience can see the show.
The equipment needs to be pushed up onto the stage. Health
and safety rules say that a ramp must have a slope of no
more than 15 degrees.
The crew need to work out how far away from the stage to
start building the ramp.
m
opp
2
adj 

 7.5
tan(x) tan(x)
Put answer in context:
The ramp must start from at least 7.5m away from the stage
From take-off, an aeroplane climbs at
an angle of 20o. When the aeroplane has
flown 10km, what height has it
reached?
km
Distance from ground.
Looking for opposite (distance from
ground) got hypotenuse.
Must be Sine formula triangle
km
Distance from ground.
Distance from ground = sin (20) x 10
= 3.42 km from the ground
A plane flies 300km on a bearing of 132 0
from an airport. How far south and east is it
from the airport. Give answer correct to 3 s.f.
North is always straight up your page for these questions..
Bearings are always measured from north around in a
clockwise direction.
Draw the problem and work out the angle at A
Then choose which side you want to find first
then choose the formula triangle to suit.
Distance South first
looking for adjacent side therefore use
cosine formula triangle
adj  cos(48)  300  201km
Distance East
looking for opposite side therefore use the
sine formula triangle
opp  sin(48)  300  223km
Ratio just means the number you
get when you divide one number by
another
Similar shapes have
the same angles –
so they have the
same angle ratios.
Sin(30) is always the
same number no
matter what size the
opp or hyp
A
Cos(30) is always the
same number no
matter what size the
adj or hyp
A1
A2
C
C1
C2
Tan(30) is always
the same number no
matter what size the
opp or adj
B
Sine ratio
Sin(x)=opp / hyp
sin(31)  0.51
opp 2.1
6


 0.51
hyp 4.1 11.74
4.3 / 8.42 = ?
To find the angle (x) when you
know sin (x) use the calculator
inverse sine function (sin-1)
• Sin (x) = 0.86
x = Sin-1 (0.86) = 59
0
• Sin (x) = 0.35
x = Sin-1 (0.35) = 21
0
Sin (x) = 0.45 , what is x ?
Sin (x) = 0.91, what is x?
27 0
66 0
Cosine ratio
cos(x) = adj / hyp
Cos(30) = 0.86
16/18.46=?
adj
4
10
cos(30) 


 0.86
hyp 4.62 11.54
To find the angle (x) when you
know cos (x) use the calculator
inverse cosine function (cos-1)
• Cos (x) = 0.86
x = Cos-1 (0.86) = 31
0
• Cos (x) = 0.35
x = Cos-1 (0.35) = 70
0
Cos (x) = 0.45 , what is x ?
Cos (x) = 0.91, what is x?
63 0
24 0
Tangent ratio
tan(x) = opp / adj
Tan(30) =0.58
7.58 / 13 = ?
opp 2.33 5.83
t an(30) 


 0.58
adj
4
10
To find the angle (x) when you
know tan (x) use the calculator
inverse tangent function (tan-1)
• tan (x) = 0.86
x tan (0.86) = 41
0
• tan (x) = 0.35
x = tan-1 (0.35) = 19
0
24 0
tan (x) = 0.45 , what is x ?
tan (x) = 0.91, what is x?
42 0
A tourist lift to the top of a cliff travels 23m from
ground to the top of the cliff. The height from
ground to the top of the cliff is 20m what is the
angle of elevation?
Sketch the problem
Choose a formula triangle
We have opposite and
hypotenuse must be sine
formula triangle
Looking for the angle BAC.
Work out the sine of the angle BAC
then use inverse sine to get angle.
20
sin( A) 
 0.869565
23
1
A  sin (0.869565)  60
o
Health and Safety stipulates that a ladder held up at the side
of a wall must have an angle of elevation between 700 and 800
to be considered safe. The height to be reached is 2.4m but
the only ladder available is 4.9m will it be classed as safe?
2.4
sin( A) 
 0.4897959
4.9
1
A  sin (0.4897959)  29
o
Not safe
Find angles A and B?
We know all sides
so ANY formula triangle will be ok to use.
6
cos( A) 
 0.832
7.21
1
A  sin (0.832)  34
o

4
tan( A)   0. 6
6
1

A  tan (0. 6)  34
o
4
cos( B) 
 0.554785
7.21
1
B  sin (0.554785)  56
o
6
tan( B)   1.5
4
1
B  tan (1.5)  56
o