Transcript trigonometric_calculations_intro

© T Madas
© T Madas
We know that:
Opp
tanq =
Adj
it is also true that:
Opp
Adj =
tanq
and:
Opp = tanq ´ Adj
How can I possibly
remember all that?
What do I need it for
anyway?
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Remember
T. O. A.
Opposite
Tangent x Adjacent
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Opposite
How tall is the tower?
Adjacent
θ
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Opposite
How tall is the tower?
Adjacent = 40 m
37°
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How tall is the tower?
O
T A
Opposite
Opposite
Tangent x Adjacent
Adjacent = 40 m
37°
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How tall Opp
is the
tower?
= Adj
x tanθ
Opp = 40 x tan37°
Opposite
Opp = 30.14 m
Adjacent = 40 m
37°
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More Tangent Calculations
O
T A
?
30°
10 m
Opposite
Tangent x Adjacent
Opp = Adj x tanθ
Opp = 10 x tan30°
Opp = 5.77 m
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More Tangent Calculations
18 m
O
T A
Opposite
?
40°
Tangent x Adjacent
Adj = Opp ÷ tanθ
Adj = 18 ÷ tan40°
Adj = 21.45 m
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The Sine
The Cosine
Opposite
Adjacent
Sine x Hypotenuse
Opposite
Cosine x Hypotenuse
Opp
sin q =
Hyp
cos q =
S O H
C A H
---
The Tangent
Tangent x Adjacent
Adj
Opp
tan q =
Adj
Hyp
---
T
O
A
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© T Madas
Sine & Cosine Calculations
15 m
O
S H
40°
Opposite
Sine x Hypotenuse
Hyp = Opp ÷ sinθ
Hyp = 15 ÷ sin40°
Hyp = 23.34 m
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Sine & Cosine Calculations
A
C H
30°
10 m
Adjacent
Cosine x Hypotenuse
Hyp = Adj ÷ cosθ
Hyp = 10 ÷ cos30°
Hyp = 11.55 m
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Sine & Cosine Calculations
A
C H
?
27°
Adjacent
Cosine x Hypotenuse
Adj = cosθ x Hyp
Adj = cos27° x 12
Adj = 10.69 m
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Sine & Cosine Calculations
O
S H
?
21°
Opposite
Sine x Hypotenuse
Opp = sinθ x Hyp
Opp = sin21° x 15
Opp = 5.38 m
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© T Madas
A cylindrical glass has a diameter of 6 cm and a straw is resting
on the glass as shown, the straw protruding above the rim of
the glass by 5 cm.
The straw forms an angle of 65° with the base of the glass.
1. Calculate the height of glass, correct to 1 decimal place.
2. Calculate the length of the straw, correct to 1 decimal place.
12.9 cm
x = tan65° x 6
x
c
SOH CAH TOA
x ≈ 12.9 cm
65°
6 cm
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A cylindrical glass has a diameter of 6 cm and a straw is resting
on the glass as shown, the straw protruding above the rim of
the glass by 5 cm.
The straw forms an angle of 65° with the base of the glass.
1. Calculate the height of glass, correct to 1 decimal place.
2. Calculate the length of the straw, correct to 1 decimal place.
y
65°
12.9 cm
We can find y by using Pythagoras Theorem
or trigonometry.
If we use Pythagoras Theorem we must
obtain the value of 12.9 to greater accuracy.
We are going to use trigonometry instead so
we can use the length of 6 cm.
6 cm
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A cylindrical glass has a diameter of 6 cm and a straw is resting
on the glass as shown, the straw protruding above the rim of
the glass by 5 cm.
The straw forms an angle of 65° with the base of the glass.
1. Calculate the height of glass, correct to 1 decimal place.
2. Calculate the length of the straw, correct to 1 decimal place.
SOH CAH TOA
y
12.9 cm
6
y =
cos65°
c
The straw is
19.2 cm long
y ≈ 14.2 cm
65°
6 cm
© T Madas
© T Madas
The diagram below shows an alley between two houses.
A ladder 3.6 m long is placed against house B as shown.
The angle between the ground and the ladder is 70°.
Calculate how wide is the alley, giving your answer correct to 3
significant figures.
B
c
A
x = cos70° x 3.6
x ≈ 0.342 x 3.6
c
SOH CAH TOA
x = 1.23 m [3 s.f.]
70°
x
1.23 m
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The diagram below shows an alley between two houses.
A different ladder is placed against house A as shown.
This ladder is 4.2 m long.
Calculate the angle the ladder makes with the wall, giving your
answer correct to the nearest degree.
4.2
A
B
= sinθ
sinθ ≈ 0.293
θ = sin-1[0.293]
1.23 m
c
θ
c
1.23
c
SOH CAH TOA
θ = 17° [nearest degree]
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© T Madas
A
43681 + 88209 = d 2
d
131890 = d 2
297 mm
D
131890 = d
c
2092 + 2972 = d 2
c
C
c
209 mm
B
c
An A4 size sheet of card is a rectangle 297 mm long by 209
mm wide.
(a) Calculate the length of its diagonal, to 3 s.f.
(b) Find the acute angle formed by the diagonal and the
longer of the two sides of the rectangle, to 2 s.f.
d ≈ 363 mm
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An A4 size sheet of card is a rectangle 297 mm long by 209
mm wide.
(a) Calculate the length of its diagonal, to 3 s.f.
(b) Find the acute angle formed by the diagonal and the
longer of the two sides of the rectangle, to 2 s.f.
B
C
Which trig ratio can we use?
209 mm
A
We can use trig to find θ
θ
297 mm
D
Why should we avoid using
a ratio which involves the
length of 363 mm?
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An A4 size sheet of card is a rectangle 297 mm long by 209
mm wide.
(a) Calculate the length of its diagonal, to 3 s.f.
(b) Find the acute angle formed by the diagonal and the
longer of the two sides of the rectangle, to 2 s.f.
A
Opp
tanθ =
adj
c
209 mm
C
35°
θ
297 mm
SOH CAH TOA
D
θ = tan-1
c
209
tanθ =
297
209
297
c
B
θ ≈ 35°
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© T Madas
ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°,
DB = 9 cm and CB = 7 cm.
Calculate to 3 significant figures:
1.
2.
The length of CD
The length of AB
3.
The size of RCBD
A
72 + x2 = 92
D
C
7 cm
B
c
x 2 = 32
c
x
x 2 = 81 – 49
x = 32
c
5.66 cm
49 + x 2 = 81
c
41°
c
By Pythagoras Theorem:
x ≈ 5.66 cm [3 s.f.]
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ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°,
DB = 9 cm and CB = 7 cm.
Calculate to 3 significant figures:
1.
2.
The length of CD
The length of AB
3.
The size of RCBD
A
SOH CAH TOA
5.66 cm
y
C
9
sin41°
c
D
y =
y =
9
0.656
c
41°
y = 13.7 cm [3 s.f.]
7 cm
B
© T Madas
ABCD is a quadrilateral with RDCB = RADB = 90°, RDAB = 40°,
DB = 9 cm and CB = 7 cm.
Calculate to 3 significant figures:
1.
2.
The length of CD
The length of AB
3.
The size of RCBD
Why should we avoid using a trig
ratio which involves the length CD ?
A
C
7
9
cosθ ≈ 0.777
θ
7 cm
θ = cos-1[0.777 ]
B
c
5.66 cm
cosθ =
c
D
c
SOH CAH TOA
41°
θ = 38.9° [3 s.f.]
© T Madas
© T Madas
The figure below shows the cross section of a barn.
AE = AB and all lengths are in metres [not to scale]
1.
Calculate the length of AB.
2.
Calculate the size of the angle ABE, giving your answer to
the nearest degree.
11
D
9
C
7.84 + 20.25 = x 2
c
B
8.2
4.5
2.82 + 4.52 = x 2
c
x
x 2 = 28.09
c
E
By Pythagoras Theorem:
x = 28.09
c
2.8
A
x = 5.3 m
lengths in metres
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The figure below shows the cross section of a barn.
AE = AB and all lengths are in metres [not to scale]
1.
Calculate the length of AB.
2.
Calculate the size of the angle ABE, giving your answer to
the nearest degree.
Since we know the exact values of
all the sides of the right angled
triangle we can use any one of the
three trigonometric ratios:
A
D
9
lengths in metres
C
sinθ =
2.8
c
11
8.2
4.5
B
5.3
sinθ ≈ 0.528
θ = sin-1[0.528]
c
θ
E
c
2.8
SOH CAH TOA
θ = 32° [nearest degree]
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© T Madas
The figure below shows a right angled trapezium ABCD.
All lengths are in cm [not to scale]
1.
Calculate the length of BC.
2.
Calculate the size of the angle DCB, giving your answer
correct to 1 decimal place.
3.3
lengths in cm
10.89 + 31.36 = x 2
C
c
c
x
x 2 = 42.25
c
6
3.32 + 5.62 = x 2
x = 42.25
c
D
By Pythagoras Theorem:
B
2.7
5.6
5.6
A
x = 6.5 cm
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The figure below shows a right angled trapezium ABCD.
All lengths are in cm [not to scale]
1.
Calculate the length of BC.
2.
Calculate the size of the angle DCB, giving your answer
correct to 1 decimal place.
6
3.3
lengths in cm
C
tanθ =
5.6
3.3
tanθ ≈ 1.697
θ = tan-1[1.697 ]
c
θ
c
SOH CAH TOA
c
D
B
2.7
5.6
5.6
A
Since we know the exact values of all the
sides of the right angled triangle we can use
any one of the three trigonometric ratios:
θ = 59.5° [1 d.p.]
© T Madas
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of D
above the level of A
3. The length EC
4. The height of the water
level if the tank was to be
stood upright with AD
horizontal.
C
E
B
D
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
C
SOH CAH TOA
x = 40 x sin40°
x = 25.7 cm [3 s.f.]
E
B
D
x
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of D
above the level of A
3. The length EC
4. The height of the water
level if the tank was to be
stood upright with AD
horizontal.
C
E
B
D
x
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of D
above the level of A
C
SOH CAH TOA
y = 30 x sin50°
y = 23.0 cm [3 s.f.]
E
B
D
x
y
50°
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of D
above the level of A
3. The length EC
4. The height of the water
level if the tank was to be
stood upright with AD
horizontal.
C
α
E
40°
B
D
x
y
50°
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
SOH CAH TOA
30
α =
0.839
c
30
tan40°
E
40°
B
c
α =
α
α = 35.8 cm [3 s.f.]
D
x
y
50°
40°
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of D
above the level of A
3. The length EC
4. The height of the water
level if the tank was to be
stood upright with AD
horizontal.
C
E
B
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
E
B
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
B
E
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
B
E
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
B
E
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
C
B
E
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
The area of the
rectangle (above the
water level) must be
equal to the area of
the triangle BCE
C
B
h
1 x
30 x 35.8 = 30 x h
2
h = 17.9 cm
E
D
A
© T Madas
The figure below shows the rectangular cross-section ABCD of a
fish tank with AB = CD = 40 cm and BD = DA = 30 cm and the side
AB tilted at an angle of 40° to the horizontal.
The water inside the tank is level with point B.
Calculate to 3 significant figures:
1. The vertical height of B
above the level of A
2. The vertical height of CD
above the level of A
3. The length EC
4. The height of the water
level if the tank was to be
stood upright with AD
horizontal.
B
h
E
D
A
© T Madas
© T Madas