Trigonometry:SIN COS TANor:
Trigonometry: SIN COS TAN or SOHCAHTOAWe will use Trigonometry to solve a number of problems:
We will use Trigonometry to solve a number of problems Measure the length of the shadow, and the angle of elevation How could you find the height of this flagpole? x SOHCAHTOAFirst identify the sides of the triangle:
First identify the sides of the triangle The longest side is called the…
Hypotenuse.
The side opposite is called the…
Opposite.
The remaining side is the…
Adjacent OPP ADJ HYP x SOHCAHTOAThe Calculations:
Suppose you measure the length of shadow to be 12 metres. This is “ADJ”
Suppose the angle is 40 OPP 40 The Calculations OPP ADJ Tan x Which trig button is this?
Hint: TOA OPP = Tan 40 12
= 0.839 12
= 10.07 m SOHCAHTOA ADJ = 12Another example :
Another example Joe buys a ladder which extends to 5 metres.
However he would not feel safe if the angle of the ladder exceeds 70 OPP 5 70 How far up the wall would the ladder extend at this angle?
What trig function is needed?
Hint:You know the HYP You need SIN SOHCAHTOAThe Calculation:
The length of ladder is 5 m; this is “HYP” The Calculation OPP HYP Sin x Which trig button is this?
Hint: SOH OPP = Sin 70 HYP
OPP = Sin 70 5
= 0.940 5
= 4.70 m 70 5 SOHCAHTOA OPPFinding an Angle:
Finding an Angle SOHCAHTOA x ADJ = 4 HYP = 5 OPP The base of this triangle is 4 cms, the hypotenuse is 5 cms.
How can you find the angle x?
Which trig button is this?
Hint: _AH Use COS…..BUT Cos x = ADJ HYP
Cos x = 4 5 = 0.8
In order to find the angle, use ”SHIFT COS” (or INV COS) x = COS-1 0.8 =36.9 ADJ HYP Cos x Another example:
Another example Sue has a ladder which reaches 3m up the wall when the angle is 59
How long is the ladder? OPP HYP 59 What trig function is needed?
Hint:You know OPP You need SIN OPP HYP Sin x HYP = OPP Sin 59
= 3 0.8572
= 3.5 m SOHCAHTOAA further example:
A further example A stepladder has the shape of an isosceles triangle. The distance between its feet is 2.2 m.
The angle the legs make with the horizontal is 64 How long are the sides of the ladder?
How high does the top reach? 2.2 m 64 SOHCAHTOA Calculations:
Calculations First you need to work with a right angled triangle. What trig button is needed? 64 B C A You need COS HYP Cos x ADJ HYP AC = AB cos 64
= 1.1 0.438 = 2.5 m How do you find the height BC? AC is the hypotenuse in ABC.
AB is the adjacent, length 1.1 m. D SOHCAHTOA OPP HYP ADJ = 1 .1 Finding the Height:
Finding the Height SOHCAHTOA OPP OPP ADJ Tan x ADJ = 1 . 1 You could use TAN. OPP = Tan 64 ADJ
= 2.050 1.1
= 2.26 m.A Final example:
A Final example The participants in a TV series are ‘dumped’ on an uninhabited island somewhere…
One of the problems they have to solve is to find the location of their island.
The first step is to find the latitude - essentially, this determines how far north (or south) you are.
This can be done by measuring the angle the North Star makes with the horizontal. (At the North Pole, it is overhead!)
It would be quite feasible to make a rudimentary protractor, but this might not be very accurate. SO... SOHCAHTOAThe Solution:
The Solution The idea is to line up the star with a suitable tall object, whose height you can measure. To keep things simple, let’s suppose you have a 4m pole. Also suppose that when you line up your eye, the North Star appears behind the top of the pole, and your eye is 432 cms from the pole as measured along the horizontal. What sides in the triangle do you know? The Opposite and Adjacent. Which Trig. Button is this? 400 cms 432 cms SOHCAHTOAThe Latitude:
The Latitude ADJ OPP Tan x SOHCAHTOA Use Tan Tan x = 400 432
= 0.9259 To find the angle x, you must do “Shift Tan” (or Tan-1) Tan-1 0.9259 = 42.8. (Your latitude is 42.8)