Trigonometry SIN COS TAN or SOHCAHTOA We will

Trigonometry: SIN COS TAN or SOHCAHTOA

We will use Trigonometry to solve a number of problems How could you find the height of this flagpole? Measure the length of the shadow, and the angle of elevation x SOHCAHTOA

First identify the sides of the triangle SOHCAHTOA The longest side is called the… Hypotenuse. The side opposite is called the… Opposite. The remaining side is the… Adjacent HYP x ADJ OPP

The Calculations SOHCAHTOA Suppose you measure the length of shadow to be 12 metres. This is “ADJ” Suppose the angle is 40 Which trig button is this? Hint: TOA OPP Tan x ADJ OPP = Tan 40 12 OPP = 0. 839 12 = 10. 07 m 40 ADJ = 12

Another example SOHCAHTOA Joe buys a ladder which extends to 5 metres. However he would not feel safe if the angle of the ladder exceeds 70 How far up the wall would the ladder extend at this angle? 5 OPP What trig function is needed? Hint: You know the HYP You need SIN 70

The Calculation SOHCAHTOA The length of ladder is 5 m; this is “HYP” OPP Which trig button is this? Hint: SOH OPP = Sin 70 HYP OPP = Sin 70 5 = 0. 940 5 = 4. 70 m Sin x HYP 5 70 OPP

Finding an Angle SOHCAHTOA 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 HYP = 5 OPP 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 x ADJ = 4 ADJ Cos x HYP

Another example SOHCAHTOA Sue has a ladder which reaches 3 m up the wall when the angle is 59 HYP How long is the ladder? 59 What trig function is needed? Hint: You know OPP You need SIN HYP = OPP Sin 59 = 3 0. 8572 = 3. 5 m OPP Sin x HYP OPP

A further example SOHCAHTOA 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

Calculations SOHCAHTOA First you need to work with a right angled triangle. C AC is the hypotenuse in ABC. AB is the adjacent, length 1. 1 m. What trig button is needed? You need COS HYP AC = AB cos 64 = 1. 1 0. 438 = 2. 5 m A HYP OPP 64 B ADJ = 1. 1 ADJ How do you find the height BC? Cos x HYP D

Finding the Height SOHCAHTOA You could use TAN. OPP Tan x ADJ OPP = Tan 64 ADJ = 2. 050 1. 1 = 2. 26 m. ADJ = 1. 1

A Final example SOHCAHTOA 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. . .

The Solution SOHCAHTOA 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 4 m 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? 400 cms The Opposite and Adjacent. Which Trig. Button is this? 432 cms

The Latitude SOHCAHTOA Use Tan OPP Tan x ADJ 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 )