Moving on from Pythagoras Theorem, we can use Trigonometry to find a missing angle of a right angle triangle, or the length of an unknown side of a right angle triangle, if the angle and the length of a side are known.90o
The FormulasIn Simple Trigonometry we use three main formulas and use the acronyms SOH CAH TOA to remember them.(SOW KA TOE WA)
SOHSine of the Angle = OppositeHypotenuseAlso shown as: -Sin = OppositeHypotenuse
SOHThis means the number you get when you divide the length of the opposite side of the triangle, by the length of the hypotenuse side53This number is the SINE of the angle
CAHCosine of the Angle = AdjacentHypotenuseAlso shown as: -Cos = AdjacentHypotenuse
CAHThis means the number you get when you divide the length of the adjacent side of the triangle, by the length of the hypotenuse side54This number is the COSINE of the angle
TOATangent of the Angle = OppositeAdjacentAlso shown as: -Tan = OppositeAdjacent
TOAThis means the number you get when you divide the length of the opposite side of the triangle, by the length of the adjacent side34This number is the TANGENT of the angle
Try transposing all three formulasSin = OppositeHypotenuseCos = AdjacentHypotenuseTan = OppositeAdjacent
The SINE, COSINE and TANGENT of the angle are not a measurement. They are a ratio made up of the lengths of two sides of the triangle.We can use this number to find the actual angle in degrees. This used to be done using tables, but is now achieved by using a calculator
All three of the SOH CAH TOA formulas are simple divisions, but the answer gives the Sine, Cosine and Tangent of the angle. To find the angle we need to use the inverse Sine, Cosine and Tangent function to find the actual angle. On a calculator these are shown as: -Cos-1Sin-1Tan-1These functions are accessed by pressing the SHIFT, or 2nd Function
It is helpful to recognise that the internal angles of triangle will always add up to 180 degrees and in a right angle triangle one angle will always be 90 degrees. Using these rules we can find any missing angle, or missing side length if there is enough information in the triangle.90o
Lets try a SIMPLE example90o4mIn this example = 30oFind the length of the Hypotenuse.
Lets try a SIMPLE example90o4mUsing the formula Cos 30o = AdjacentHypotenuseWe can transpose to find: -Hypotenuse = 4.62m
There are two ways to find the answer.Using Pythagoras TheoremUsing Trigonometry Formulas90o4mNow find the length of the opposite side4.62m
Pythagoras tells us: -
c = a2 b290o4mNow find the length of the opposite side4.62mc = (4.622 42)c = 2.31m
We can also say: -opposite = hypotenuse x Sin 30o90o4mNow find the length of the opposite side4.62mopposite = 4.62 x Sin 30oopposite = 2.31m
Deciding which of the trigonometry formulas to use may seem complicated, but look at the problem and see what information you have been given. If you have measurements for the opposite and hypotenuse side, use SOH. If you have and the adjacent side you can use CAH, or TOA to find the answer.
Try This One90o30ma = ?mc = ?mIf = 39.64o Find the length of the hypotenuse and opposite sides.opposite = hypotenuse x Sin
Try This One90o39.64o30m39.05m25mBased on what you have learned already what is the value of the angle marked
Just One More90o125mma = ?m95mmFor this example find the length of the hypotenuse and the angle marked as a2 = b2 + c2a2 = 1252 + 952a = 1252 + 952a = 157mm
Just One More90o125mm157mm95mmFor this example find the angle marked as Cos -1 = 37.23o
Last One I Promise!!!90o36.87o125mm157mm95mmWhat is the value of the angle marked ?