Trigonometry

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Trigonometry

• Right angled triangles

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A triangle
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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

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SOH CAH TOASine, Cosine Tangent of an angle
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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)

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TOA SOH CAH

• TOA
• Toms 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

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SOH CAH TOA

• SOH

S
O
H
Opposite side sin (x) times hypotenuse
Hypotenuse opposite side divided by sin (x)
Sin (x) opposite side divided by hypotenuse
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SOH CAH TOA
C
A

• CAH

H
Adjacent side cos (x) times hypotenuse
Hypotenuse adjacent side divided by cos (x)
Cos (x) adjacent side divided by hypotenuse
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SOH CAH TOA
A

• TOA

T
O
Opposite side tan (x) times adjacent side
Adjacent side opposite side divided by tan (x)
tan (x) opposite side divided by adjacent side
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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
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• 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
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Put answer in context The ramp must start from
at least 7.5m away from the stage
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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.
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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
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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.
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Draw the problem and work out the angle at AThen
choose which side you want to find first then
choose the formula triangle to suit.
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Distance South firstlooking for adjacent side
therefore use cosine formula triangle
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Distance East looking for opposite side
therefore use the sine formula triangle
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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
Tan(30) is always the same number no matter what
size the opp or adj
B
C1
C2
C
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Sine ratioSin(x)opp / hyp
4.3 / 8.42 ?
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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
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Cosine ratiocos(x) adj / hyp
Cos(30) 0.86
16/18.46?
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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
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Tangent ratiotan(x) opp / adj
Tan(30) 0.58
7.58 / 13 ?
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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
• tan (x) 0.45 , what is x ?
• tan (x) 0.91, what is x?

24 0
42 0
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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?
Choose a formula triangle
Sketch the problem
We have opposite and hypotenuse must be sine
formula triangle
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Looking for the angle BAC.Work out the sine of
the angle BAC then use inverse sine to get angle.
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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?
Not safe
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Find angles A and B?We know all sides so ANY
formula triangle will be ok to use.
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