Transcript Introduction to trigonometry

Introduction to trigonometry
M. S. Martin
Feb 2007
Basic trigonometry equations
• SIN =
OPP
HYP
HYPOTENUSE
OPPOSITE
deg°
ADJACENT
• COS = ADJ
HYP
• TAN = OPP
ADJ
• Remember SOH,
CAH, TOA
Now lets look at the
calculator
• Find these functions
on the calculator.
• Sin, Cos, Tan
Now lets look at how we use the
equations
HYPOTENUSE
OPPOSITE
?
28°
ADJACENT
3.600m
• We will use an example of 28°
and the right angle triangle
opposite
• We know the degrees, we know
the length of one side, but not the
length of the opposite side ?
• So we can apply the ‘tan’ formula
here
• Tan 28° = OPP or ?
ADJ
3.6
• We have to re-arrange the formula
• OPP = Tan 28° x 3.600
• Tan 28° x 3.600 = 1.914
• Now lets just look at that math's
Re-arrangement of the equation
• Tan 28° = OPP or ?
ADJ
3.6
• Same as, Tan 28° X
= OPP
1
3.600
• Apply cross multiplication
• 1 x opp = tan 28° x 3.600
• opp = tan 28° x 3.600
= 1.914
• An easy way to remember
for rearranging other
formula’s is
• Think of a simple equation
4
=
8
2
• Find the corresponding
number to your formula and
look at what you have to do
to the other numbers
• Meaning, you want 8 so
4x2=8
• If you wanted 2, 8 ÷ 4 = 2
Write the equation for the x° in the
following triangles
7
5
9
6
X°
X°
8
8
6
Tan x = 8
Sin x = 6
9
Cos x = 8
9
5
Tan x = 7
Sin x = 5
8
Cos x = 7
8
Write the equation, then solve it
?
50m
?
25°
30°
Sin 30° = OPP
HYP
Sin 30° = OPP
50
Re-arrange
OPP = Sin 30° x 50
= 25
14.670m
Cos 25° = ADJ
HYP
Cos 25° = 14.670
HYP
Re – arrange
HYP
= 14.670
Cos 25°
= 16.187
Now let’s look at how it
relates to a building
Firstly to calculate the drop off
measurement in a roof frame
Drop off measurements
• Can you see the drop
off measurement ?
• Can you see the
relevent triangle ?
Handout, Now lets
look at a example
Use ‘Tan’ to calculate drop off
• Tan 30° = opposite side
adjacent side
• Tan 30° = opposite side
400 + 110 + 40
• Tan 30° = opposite side
550
• Re-arrange formula
• Opp side = tan 30° x 550
= 317mm
• Total drop off = 317 + 20
= 337mm
Pitch and eave width will change
the drop off – see below
Same Roof Pitch
30°
Drop-off
Same Roof Pitch
30°
Drop-off
Smaller eaves width
Larger eaves width
Smaller Drop-off
(Higher Wall)
Larger Roof Pitch
30°
Larger Drop-off
(Lower Wall)
Smaller Roof Pitch
22½°
Smaller eaves width
Larger eaves width
SAME WALL HEIGHT
Same Wall Height
Exercises
• Calculate the drop off,
assume 20mm for
birdsmouth cut.
Tan 26° = opposite side
400 + 110 + 40
Tan 26° = opposite side
550
26°
Re-arrange formula
400
Opp side = tan 26° x 550
= 268mm
Total drop off = 268 + 20
110, 40, 100
=
288 mm
Exercises
• Calculate the drop off,
assume 20mm for
birdsmouth cut.
Tan 22° = opposite side
350 + 110 + 50
Tan 22° = opposite side
510
Re-arrange formula
Opp side = tan 22° x 510
= 206 mm
Total drop off = 206 + 20
= 226 mm
22°
350
110, 50, 90
Exercises
• Calculate the drop off,
assume 20mm for
birdsmouth cut.
Tan 36° = opposite side
300 + 110 + 60
Tan 36° = opposite side
470
Re-arrange formula
Opp side = tan 36° x 470
= 341 mm
Total drop off = 341 + 20
= 361 mm
36°
300
110, 60, 100
Exercises
• Calculate the drop off,
assume 20mm for
birdsmouth cut.
Tan 32° = opposite side
250+ 110 + 40
Tan 32° = opposite side
400
Re-arrange formula
Opp side = tan 32° x 400
= 250 mm
Total drop off = 250 + 20
= 270 mm
32°
250
110, 40, 90