2 Trigonometric RatiosA RATIO is a comparison of two numbers. For example;boys to girlscats : dogsright : wrong.In Trigonometry, the comparison is betweensides of a triangle.
3 Finding Trig RatiosA trigonometric ratio is a ratio of the lengths of two sides of a right triangle.The word trigonometry is derived from the ancient Greek language and means measurement of triangles.The three basic trigonometric ratios are called sine, cosine, and tangent
5 Some terminology:Before we can use the ratios we need to get a few terms straightThe hypotenuse (hyp) is the longest side of the triangle – it never changesThe opposite (opp) is the side directly across from the angle you are consideringThe adjacent (adj) is the side right beside the angle you are considering
6 A picture always helps… looking at the triangle in terms of angle bbA is the adjacent (near the angle)CAB is the opposite (across from the angle)BbNearC is always the hypotenusehypLongestadjoppAcross
7 But if we switch angles… looking at the triangle in terms of angle aA is the opposite (across from the angle)CAaB is the adjacent (near the angle)BAcrossC is always the hypotenusehypLongestoppaadjNear
9 One more thing… θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’.Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized.
10 A hypotenuse c b leg SINE COSINE TANGENT B leg a In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuseAWe’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.adjacentoppositehypotenusecFirst let’s look at the three basic functions.blegSINECOSINETANGENTBlegaThey are abbreviated using their first 3 letters
11 INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE the trig functions of the angle B using the definitions.ASOHCAHTOASOHCAHTOAhypotenusecINEbOSINEANGENToppositePPOSITEDJACENTYPOTENUSEPPOSITEDJACENTYPOTENUSEadjacentBa
12 It is important to note WHICH angle you are talking about when you find the value of the trig function.ALet's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem sohypotenuse5cb4oppositeoppositeadjacentBtan B =a3sin A =SOH-CAH-TOA
13 You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.Oh, I'm acute!AThis method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.54So am I!B3
14 SOH CAH TOA O S H A C H O T A One more time… Here are the ratios: sinθ = opposite sidehypotenuseHACcosθ = adjacent sidehypotenuseHOTtanθ =opposite sideadjacent sideASOH CAH TOA
15 The nice thing is that your calculator has a tan, sin and cos key that can save you some work.However, you must remember to change the settings on your calculator from Radians to Degrees. Radians is the default setting.Here are the steps:Press ModeMove the curser down to the 3rd line “Radians”Slide the curser over so “degree” is blinkingPress 2nd Quit
16 Make sure you have a calculator… GivenRatio of sidesAngle, sideLooking forAngle measureMissing sideUseSIN-1COS-1TAN-1SIN, COS, TAN
17 Calculating a side if you know the angle How can we use these?Calculating a side if you know the angleyou know an angle (25°) and its adjacent sidewe want to know the opposite sideACB = 625°b
18 Another example If you know an angle and its opposite side, you can find the adjacent side.A = 6CB25°b
19 How can we use it? Suppose we want to find an angle and we only know two side lengths Suppose we want to find angle aIs side A opposite or adjacent?what is side B?with opposite and adjacent we use the…A = 3CB = 4abthe oppositethe adjacenttan ratio
20 Lets solve itA = 3CB = 4abWhen the tan, sin or cos is in the denominator, we are going to use the reciprocal buttons.Look above tan on the calculatorYou should see TAN-1Press 2nd TAN (.75)
21 Another tangent example… we want to find angle bB is the oppositeA is the adjacentso we use tanbCA = 3aB = 4
22 Ex. 6: Indirect Measurement You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
23 The math The tree is about 76 feet tall. Write the ratio tan 59° =oppositeadjacentWrite the ratiotan 59° =h45Substitute valuesMultiply each side by 4545 tan 59° = hUse a calculator or table to find tan 59°45 (1.6643) ≈ hSimplify75.9 ≈ hThe tree is about 76 feet tall.
24 Ex. 7: Estimating Distance Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.30°
25 Now the mathsin 30° =oppositehypotenuse30°Write the ratio for sine of 30°sin 30° =76dSubstitute values.d sin 30° = 76Multiply each side by d.sin 30°76d =Divide each side by sin 30°0.576d =Substitute 0.5 for sin 30°d = 152SimplifyA person travels 152 feet on the escalator stairs.
28 Why do we need the sin & cos? We use sin and cos when we need to work with the hypotenuseif you noticed, the tan formula does not have the hypotenuse in it.so we need different formulas to do this worksin and cos are the ones!bC = 10A25°B
29 Lets do sin first we want to find angle a since we have opp and hyp we use sinbC = 10A = 5aB
30 And one more sin example find the length of side AWe have the angle and the hyp, and we need the oppbC = 20A25°B
31 And finally cos We use cos when we need to work with the hyp and adj so lets find angle bbC = 10A = 4aB
32 Here is an example Spike wants to ride down a steel beam The beam is 5m long and is leaning against a tree at an angle of 65° to the groundHis friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospitalHow high up is he?
33 How do we know which formula to use??? Well, what are we working with?We have an angleWe have hypWe need oppWith these things we will use the sin formulaC = 5B65°
34 So lets calculateC = 5Bso Spike will have fallen 4.53m65°
35 One last example…Lucretia drops her walkman off the Leaning Tower of Pisa when she visits ItalyIt falls to the ground 2 meters from the base of the towerIf the tower is at an angle of 88° to the ground, how far did it fall?
36 First draw a triangle What parts do we have? We have an angle We have the AdjacentWe need the oppositeSince we are working with the adj and opp, we will use the tan formulaB88°2m
37 So lets calculateBLucretia’s walkman fell 57.27m88°2m
39 An applicationYou look up at an angle of 65° at the top of a tree that is 10m awaythe distance to the tree is the adjacent sidethe height of the tree is the opposite side65°10m
40 What are the steps for doing one of these questions? Make a diagram if neededDetermine which angle you are working withLabel the sides you are working withDecide which formula fits the sidesSubstitute the values into the formulaSolve the equation for the unknown valueDoes the answer make sense?
43 Two Triangle ProblemsAlthough there are two triangles, you only need to solve one at a timeThe big thing is to analyze the system to understand what you are being givenConsider the following problem:You are standing on the roof of one building looking at another building, and need to find the height of both buildings.
44 Draw a diagramYou can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m60°40°45m
45 Break the problem into two triangles. The first triangle:The second trianglenote that they share a side 45m longa and b are heights!a60°45m40°b
46 The First TriangleWe are dealing with an angle, the opposite and the adjacentthis gives us Tana60°45m
47 The second triangleWe are dealing with an angle, the opposite and the adjacentthis gives us Tan45m40°b
48 What does it mean? Look at the diagram now: the short building is 37.76m tallthe tall building is 77.94m plus 37.76m tall, which equals m tall77.94m60°40°37.76m45m
49 Ex: 5 Using a CalculatorYou can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.