# Math 416 Trigonometry. Time Frame 1) Pythagoras 1) Pythagoras 2) Triangle Structure 2) Triangle Structure 3) Trig Ratios 3) Trig Ratios 4) Trig Calculators

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• Math 416Trigonometry

• Time Frame1) Pythagoras2) Triangle Structure3) Trig Ratios4) Trig Calculators5) Trig Calculations6) Finding the angle7) Triangle Constructions8) Word Problems

• Right Angle TrianglesThe next section will deal exclusively with right angle triangles. We recall

yxzPythagoras x2 = y2 + z2Angle Sum+ + = 180

• Pythagoras Examplex3772 = x2 + 3249 = x2 + 940 = x26.32 = x x = 6.32Do Stencil #1

• Triangle StructureWe all need to agree on what we are talking about. ConsiderbBCAca
• Trig RatiosWhen we consider the similarity of right angle triangles as long as we ignore decimal angles there are only 45 right angle triangles. Consider the angles90 1 8990 2 8890 3 87 90 45 - 45Then we start over

• Trig RatiosFrom ancient times, people have looked at the ratios within right angles trianglesFirst in tablesNow stored in calculatorsWe need to define the parts of a right angle triangleTwo types of definitions

• DefinitionsAbsolute never changesRelative involves the position

• Absolute vs Relative

cCabBANow we can define absolutely the hypotenuse as the side opposite the right angle (longest side). In this example it is side AC or b.

Now relative to angle we define side AB or c as the opposite sideNow relative to angle we define side BC or a as the opposite side

• Absolute Vs. RelativeNow relative to angle we define side BC or a as the adjacent sideNow relative to angle we define side AB or c as the adjacent side

• Labeling the TriangleHence with respect to HypAdjOppNow we define the three main trig ratios

• Trig RatiosThe sine of an angle is defined as the ratio of the opposite to the hypotenuse. Thus Sin = Opp HypThe cosine of an angle is defined as the ratio of the adjacent to the hypotenuse. Thus Cos = Adj Hyp

• Trig RatiosThe tangent of an angle is defined as the ratio of the opposite to the adjacent. Thus Tan = Opp Adj

• SOH CAH - TOAYou may of heard the acronym SOH CAH TOA or SOCK A TOASin Opp HypCos Adj HypTan Opp Adj

• Old Harry And His Old AuntThere is another acronym old Harry and his old auntSin Opp HypCos Adj HypTan Opp AdjUse the acronym that you can remember

• ExampleConsider39C1536Sin A = 15 39ABTan C = 36 15Cos C = 15 39Cos A = 36 39Tan A = 15 36Sin C = 36 39

• Trig CalculatorNow note the table for the assignment is as follows (question #3). For example

40C3224BA#Angle Sin Cos Tan Angle Sin Cos TanEg B324024403224C

244032402432

• Trig CalculatorWe note that these ratios are stored by angle albeit as decimals in a calculatorNote first and foremost your calculatorsIT MUST BE IN DEGREESMake sure you find your DRG (Degree Radian Gradients)

• Trig CalculatorHence if = 54 then to 4 decimal placesSin 54 =

0.8090Cos 54 =0.5878

Tan 54 =1.3764Do Stencil #3

• Question #4The table required for #4 is as followsExample = 37# SinCosTanEg0.60180.75360.7986

• Trig CalculationsThere are three basic type of questions. We will focus on the Sine ratio (like question #5) but the techniques are the same for all trig ratio problems.

• Trig CalculationsConsiderx1240Solve for xUse the angle given to you! Step #1: Determine the Trig Ratio involved with respect to the angle12 = hypotenuse, x = opp Thus, SINE

• Trig Calculationsx1240Step #2 Determine the equationX = sin 4012Step #3: Cross multiply (if necessary)x = 12 Sin 40

• Trig Calculationsx1240Step #4 If the unknown is isolated (by itself) solve if not divide then solvex = 7.71

• Trig CalculationsMore Practice

x3911x = sin 3911x = 11 sin 39x = 6.92

• Trig CalculationsEven More Practice

1142x11 = sin 42 x11 = x sin 42Divided both sides by sin 42 or 0.67x = 16.44

• Trig CalculationsEven More Practice

973x9 = sin 73xx = 9 . sin 73x = 9.41

• Finding the AngleUp until now we have the angle get the ratioNow we need to go the other wayGiven the ratio, give the angleEg. The buttons we are looking for are the inverse sine (sin -1)Inverse cosine (cos -1)Inverse tangent (tan -1)Find it on your calculator

• Examples of Finding the AngleFind the angle516Sin = 5 16= sin -1 ( 5 ) 16= 18 (no decimals)

• Another Example Find the angle731sin = 7 31

= 13

• Other ExamplesNow all the Trig Calculations can follow these procedures2515x = cos 25 15x = 13.59

xSin, Cos or Tan?

• Another Example Find the Sidex616x = Tan 61 6x = 6 tan 61x = 10.82

• Another Example Find the angle731sin = 7 31sin ( 7 ) 31= 13

• Another Example Find the angle57cos = 5 7 = 44

• Another Example Find the angle518tan = 18 5 = 74

• Completing the TriangleNow using our knowledge we can complete triangles

51476yxDraw this triangle and another one right below fill out missing info5 = cos 14xx = 5.15y = tan 145y = 1.25